This section details the validation work carried out in Caelus 4.10 for several conditions ranging from attached to separated flows. Both transient and steady state conditions have been considered under laminar and turbulent flows for relevant cases.

*Laminar, incompressible flow over a two-dimensional Sharp-Leading Edge flat-plate*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(e\) (subscript) | Boundary layer edge conditions | |

\(L\) | Length of the plate | \(m\) |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re\) | Reynolds number | Non-dimensional |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\eta\) | Blasius parameter | Non-dimensional |

**Introduction**

In this validation case, steady, incompressible, laminar flow over a two-dimensional sharp-leading edge flat-plate at zero angle of incidence is investigated. The flow generates a laminar boundary layer and the computational results are compared with the Blasius solution for the incompressible flow. Validation of the flow over a flat-plate forms the basis of these validation efforts as it is perhaps the most well understood configuration for CFD code-validation.

Blasius, in his work [6] obtained the solution to the Boundary Layer Equations using a transformation technique. Here, equations of continuity and momentum in two-dimensional form are converted into a single ordinary differential equation (ODE). The solution to this ODE can be numerically obtained and is regarded as the exact solution to the boundary layer equations. It is only valid for steady, incompressible, laminar flow over a flat-plate. One of the highlights of Blasius solution is the analytical expression for the skin friction coefficient (\(c_f\)) distribution along the flat-plate given by

\[c_f \approx \frac{0.644}{\sqrt{Re_{x}}}\]

where \(Re_{x}\) is the local Reynolds number defined as

\[Re_x = \frac{Ux}{\nu}\rm{,}\]

\(U\) is the freestream velocity, \(x\) is the distance starting from the leading edge and \(\nu\) is the kinematic viscosity.

**Problem definition**

This exercise is based on the validation work carried out by the NASA NPARC alliance for a flow over a flat-plate using the same conditions in the incompressible limit. A schematic of the geometric configuration is shown in Fig. 1.

The length of the plate is L = 0.3048 m wherein, x = 0 is the leading edge, the Reynolds number of the flow based on the length of the plate is 200,000 and U is the velocity in the x-direction. Assuming the inlet flow is at a temperature of 300 K, the kinematic viscosity can be determined from dynamic viscosity and density of the fluid. The value is given in the table below. The value of dynamic viscosity is obtained from the Sutherland viscosity formulation. Using the Reynolds number, plate length and kinematic viscosity, the freestream velocity evaluates to U = 10.4306 m/s. The following table summarises the freestream conditions used:

Fluid | \(Re\) | \(U~(m/s)\) | \(p~(m^2/s^2)\) | \(T~(K)\) | \(\nu~(m^2/s)\) |
---|---|---|---|---|---|

Air | 200,000 | 10.43064 | \((0)\) Gauge | 300 | \(1.58963\times10^{-5}\) |

As we have assumed the flow incompressible, the density (\(\rho\)) remains constant. In addition, since the fluid temperature is not considered, the viscosity remains constant. For incompressible flows, the kinematic forms of pressure and viscosity are always used in Caelus.

**Computational Domain and Boundary Conditions**

The computational domain is a rectangular block encompassing the flat-plate. Fig. 2 shows the details of the boundaries used in two-dimensions (\(x-y\) plane). As indicated in blue, the region of interest extends between \(0\leq x \leq 0.3048~m\) and has a no-slip boundary condition. Upstream of the leading edge, a slip boundary is used to simulate freestream uniform flow approaching the flat-plate. However, downstream of the plate, there is additional no-slip wall a further three plate lengths (highlighted in green). This ensures that the boundary layer in the vicinity of the trailing edge is not influenced by the outlet boundary. Since the flow is subsonic, disturbances cause the pressure to propagate both upstream and downstream. Therefore, placement of the inlet and outlet boundaries were chosen to have minimal effect on the solution. The inlet boundary as shown in the figure below is placed at start of the slip-wall (\(x = -0.06~m\)) and the outlet at the end of the second no-slip wall (\(x = 1.2192~m\)). Both inlet and outlet boundaries are between \(0\leq y \leq 0.15~m\). A slip-wall condition is used for the entire top boundary.

*Boundary Conditions and Initialisation*

The following are the boundary condition details used for the computational domain:

- Inlet
- Velocity: Fixed uniform velocity \(u = 10.4306~m/s\) in \(x\) direction
- Pressure: Zero gradient

- Slip wall
- Velocity: Slip
- Pressure: Slip

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)

- Initialisation
- Velocity: Fixed uniform velocity \(u = 10.4306~m/s\) in \(x\) direction
- Pressure: Zero Gauge pressure

**Computational Grid**

A 2D structured grid was generated using Pointwise in the \(x-y\) plane. Since Caelus is a 3D computational framework, it necessitates the grid to also be 3D. Therefore, a 3D grid was obtained using Pointwise by extruding the 2D grid in the positive \(z\) direction by *one cell*. The final 3D grid was then exported to the Caelus format (polyMesh). The two \(x-y\) planes obtained as a result of grid extrusion need boundary conditions to be specified. As the flow over a flat-plate is generally 2D, we do not need to solve the flow in the third dimension. This is achieved in Caelus by specifying *empty* boundary condition for each plane. Although, no flow is computed in the \(z\) direction, a velocity of \(w = 0\) has to be specified for the velocity boundary condition as indicated above.

Fig. 3 shows the 2D grid in the \(x-y\) plane. As can be seen, the grid is refined perpendicular to the wall in order to capture resolve the viscous effects. To ensure that the gradients in boundary layer are well resolved, about 50 grid nodes are placed between the wall and the boundary layer edge. Grid refinement is also added at the leading edge so that the growth of the boundary layer is also well resolved. In this particular case, 400 cells were used in the stream-wise (\(x\)) direction (\(x \leq 0 \leq 0.3048~m\)) and 600 in the wall normal (\(y\)) direction. For no-slip wall beyond \(x > 0.3048\), a similar distribution is used.

**Results and Discussion**

A time-dependent solution to the two-dimensional flat-plate was obtained using Caelus 4.10. The SLIM transient solver was used here and the flow was simulated sufficiently long (several plate length flow times) such that steady flow was established. For the discretization of time-dependent terms, the first-order Euler scheme was used. A Gauss linear discretization was used for the pressure and velocity gradients. A linear upwind discretization was for the divergence of velocity and mass flux. A linear corrected scheme was used for Laplacian discretization while cell-to-face centre interpolation used linear interpolation.

In Fig. 4, the skin-friction distribution along the flat-plate obtained from the CFD simulation is compared with that of the Blasius analytical solution. Here, the distance \(x\) is normalised with the length of the plate (\(L\)). Excellent agreement is observed along the entire length of the flat-plate.

At the exit plane of the flat-plate at \(x = 0.3048~m\), velocity data was extracted across the boundary layer and compared with the Blasius analytical solution. This is shown in Fig. 5 where the velocity profile is plotted using similarity variables from the Blasius solution. Here, \(\eta\) is the non-dimensional distance from the wall to the boundary layer edge and \(U_e\) is the velocity at the boundary layer edge. Similar to skin-friction, the velocity profile also exhibits excellent agreement with the Blasius solution.

**Conclusions**

The steady, incompressible, laminar flow over a two-dimensional flat-plate was simulated using Caelus 4.10 utilising the SLIM solver. The results were validated against the Blasius analytical solutions resulting in excellent agreement.

*Laminar, incompressible flow in a 90 degree tee junction*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(L\) | Length of the branch | \(m\) |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_w\) | Reynolds number based on width | Non-dimensional |

\(V\) | Freestream velocity in y-direction | \(m/s\) |

\(W\) | Width of the branch | \(m\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

**Introduction**

In this validation case, laminar, incompressible flow through a two-dimensional \(90^\circ\) tee junction was investigated. Due to the presence of the side branch, the flow separates and forms a recirculating region. The recirculating regions influences the mass flow through the main and side branches. The numerically computed mass-flow ratio was calculated and compared with experiment.

A comprehensive study of flow through planar branches has been carried out by Hayes et al. [12] due to its prevelance in the bio-mechanical industry. Of which, the \(90^\circ\) right-angled tee junction is considered here for the purpose of validation.

**Problem definition**

The following figure shows the schematic of the tee-junction. Here, L = 3.0 m, W = 1.0 m respectively, the Reynolds number based on the width is 300, and V is the velocity in the y-direction. For simplicity, we have assumed the velocity, V = 1 m/s. Using these values the resulting kinematic viscosity was 0.00333 \(m^2/s\).

The summary of the flow properties and geometric details are given in the following table.

\(Re_w\) | \(L\) | \(W\) | \(V~(m/s)\) | \(p~(m^2/s^2)\) | \(\nu~(m^2/s)\) |
---|---|---|---|---|---|

300 | 3.0 | 1.0 | 1.0 | \((0)\) Gauge | 0.00333 |

As we have assumed the flow incompressible, the density (\(\rho\)) remains constant. In addition, since the fluid temperature is not considered, the viscosity remains constant. For incompressible flows, the kinematic forms of pressure and viscosity are always used in Caelus.

**Computational Domain and Boundary Conditions**

Since this is an internal flow problem, the computational domain is contained within tee-junction geometry. The details are shown in Fig. 7. As indicated, all tee-junction walls have a no-slip boundary condition which has been highlighted in blue. At the inlet, a fully developed laminar flow parabolic profile is applied, otherwise a much longer main branch would be required for the flow to develop. The domain has two outlets, one at the end of the main channel and the other at the end of side branch. Note the exit pressures at the two outlets are equal.

*Boundary Conditions and Initialisation*

The following are the boundary condition details used for the computational domain:

- Inlet
- Velocity: Parabolic velocity profile; centerline velocity of \(v = 1.0~m/s\) in \(y\) direction
- Pressure: Zero gradient

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient

- Outlet-1
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)

- Outlet-2
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)

- Initialisation
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero Gauge pressure

**Computational Grid**

The 2D structured grid is shown in Fig. 8. Since Caelus is a 3D computational framework, it necessitates the grid to also be 3D. Therefore, a 3D grid was obtained by extruding the 2D grid in the positive \(z\) direction by *one cell*. The final 3D grid was then exported to the Caelus format (polyMesh). The two \(x-y\) planes obtained as a result of grid extrusion need boundary conditions to be specified. As the flow over a flat-plate is generally 2D, we do not need to solve the flow in the third dimension. This is achieved in Caelus by specifying *empty* boundary condition for each plane. Although, no flow is computed in the \(z\) direction, a velocity of \(w = 0\) has to be specified for the velocity boundary condition as indicated above.

A total of 2025 cells comprise the tee-junction of which, 90 cells are distributed along the height of the main channel, and 45 along the length of the side branch. The distribution is such that a dimensional length of \(L = 1~m\) has a total of 45 cells, giving a distribution of \((2/3)*45 = 30\) cells for the \((2/3) L\) segment of the main channel. The width, \(W\), consists of 15 cells.

**Results and Discussion**

A time-dependent solution to the two-dimensional flat-plate was obtained using Caelus 4.10. The SLIM transient solver was used here and the flow was simulated sufficiently long such that steady separated flow was established. To ensure this, shear-stress distribution was monitored on the lower wall of the side branch. The simulation was stopped once the separation and reattachment locations no longer varied with time.

Mass flow was calculated at the inlet and at the main outlet (outlet-1) and the mass-flow ratio was subsequently calculated. The below table compares the SLIM result with the experimental value. As can be noted, the agreement between the two is excellent.

Experimental | SLIM | Percentage Difference | |
---|---|---|---|

Flow Split | \(0.887\) | \(0.886\) | \(0.112~\%\) |

**Conclusions**

The steady, incompressible, two-dimensional laminar flow in a right-angled \(90^\circ\) tee-junction was simulated using Caelus 4.10 with the SLIM solver and validated against experimental data resulting in excellent agreement.

*Laminar, incompressible flow over a circular cylinder*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(D\) | Diameter of the cylinder | \(m\) |

\(f\) | Frequency | \(hz\) |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_D\) | Reynolds number based on diameter | Non-dimensional |

\(St\) | Strouhal number | Non-dimensional |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

In this validation study, laminar incompressible flow over a 2D circular cylinder is investigated at a Reynolds number of 100. This classical configuration represents flow over a bluff body dominated by a wake region. For flows having a low Reynolds number (\(40 \leq Re_D \leq 150\)), periodic vortex shedding occurs in the wake. The phenomenon of vortex shedding behind bluff bodies is referred to as the *Karman Vortex Street* [2] and provides an transient case for CFD code validation.

In his work, Roshko [2] experimentally studied wake development behind two-dimensional circular cylinders from Reynolds number ranging from 40 to 10000. For Reynolds numbers of 40 to 150, the so called the *stable range* [2], regular vortex streets are formed with no evidence of turbulence motion in the wake. Therefore, at a Reynolds number of 100, the vortex shedding exhibits smooth, coherent structures making it ideally suited for validating laminar CFD calculations. The frequency associated with the oscillations of the vortex streets can be characterized by the Strouhal Number (\(St\)). The Strouhal Number is a non-dimensional number defined as

\[St = \frac{fD}{U}\]

where, \(f\) is the frequency of oscillations of vortex shedding, \(D\) is the diameter of the cylinder and \(U\) is the freestream velocity of the flow. Experimentally [2], it has been determined that for a Reynolds number based on the diameter of the cylinder of 100, the Strouhal number \(St \approx 0.16 - 0.17\). One of the main objectives in this study was to compare the \(St\) for the CFD calculation to the experimental data of Roshko [2]. Provided the cylinder has a sufficient span length, the flow characteristics can be assumed to be two-dimensional as the experiments suggest.

**Problem Definition**

Fig. 9 shows the schematic of the two-dimensional circular cylinder. Here, the diameter D = 2 m and is the characteristic length for the Reynolds number, which is 100. For simplicity, the freestream velocity was taken to be U = 1 m/s in the x direction. Using these values the kinematic viscosity was calculated to be 0.02 \(m^2/s\).

In the table below, a summary of the freestream conditions are provided

\(Re_D\) | \(D\) | \(U~(m/s)\) | \(p~(m^2/s^2)\) | \(\nu~(m^2/s)\) |
---|---|---|---|---|

100 | 2.0 | 1.0 | \((0)\) Gauge | 0.02 |

Here, the flow is assumed to be incompressible and therefore the density is constant. Further, no temperature is evaluated in this calculation and hence the viscosity also remains constant. For incompressible flows, the kinematic forms of pressure and viscosity are always used in Caelus.

**Computational Domain and Boundary Conditions**

A rectangular computational domain in the \(x-y\) plane was constructed surrounding the circular cylinder as shown in Fig. 10. A full cylinder must be used due to the oscillatory nature of the shed vortices. Rhe domain extends by 5 diameters of cylinder and 20 diameters downstream. Since the flow here is viscous dominated, sufficient downstream length is required to capture the vortex separation from the surface of the cylinder and the subsequently shedding in the wake. In the \(y\) direction, the domain extends 5 diameters on either side. From the figure, multiple inlet boundaries to this domain can be seen, one at the upstream boundary and the other two for the top and bottom boundaries. This type of configuration is needed to appropriately model the inflow, similar to an undisturbed flow in an experimental set-up. It is noted that for top and bottom boundaries, the flow is in the \(x\) direction. The outlet is located at the downstream boundary. The cylindrical wall is a no-slip boundary condition.

*Boundary Conditions and Initialisation*

Following are the details of the boundary conditions used:

- Inlet-1
- Velocity: Fixed uniform velocity \(u = 1.0~m/s\) in \(x\) direction
- Pressure: Zero gradient

- Inlet-2
- Velocity: Fixed uniform velocity \(u = 1.0~m/s\) in \(x\) direction
- Pressure; Zero gradient

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)

- Initialisation
- Velocity: Fixed uniform velocity \(u = 1.0~m/s\) in \(x\) direction
- Pressure: Zero Gauge pressure

**Computational Grid**

The computational grid in 2D was generated using Pointwise in the \(x-y\) plane. Since Caelus is a 3D computational framework, it necessitates the grid to also be 3D. Therefore, a 3D grid was obtained using Pointwise by extruding the 2D grid in the positive \(z\) direction by *one cell*. The final 3D grid was then exported to the Caelus format (polyMesh). The two \(x-y\) planes obtained as a result of grid extrusion need boundary conditions to be specified. As the flow over a flat-plate is generally 2D, we do not need to solve the flow in the third dimension. This is achieved in Caelus by specifying *empty* boundary condition for each plane. Although, no flow is computed in the \(z\) direction, a velocity of \(w = 0\) has to be specified for the velocity boundary condition as indicated above.

The 2D domain consisted of 9260 cells. An O-grid topology was constructed around the cylinder (see the right figure) with 10 cells in the radial direction and 84 cells in the circumferential direction. 31 cells were used upstream of the O-grid, in the \(x\) direction while 100 cells were used downstream. The region of interest is about 10 diameters downstream, where the grids are refined. In the \(y\) direction, 21 cells were used above and below the O-grid region.

**Results and Discussion**

A time-dependent simulation was carried out using the Caelus 4.10 with the SLIM solver. To capture the transient start-up process, the calculation was started from time t = 0 s and was simulated up to t = 360 s, while lift and drag forces over the cylindrical surface were monitored at a frequency of 2 Hz. It was found that the on-set of vortex shedding occurred after about t = 90 s which was then followed by a steady shedding process. A Fast Fourier transformation (FFT) was carried out on the lift force data and the peak frequency of vortex shedding occurred at \(f = 0.0888\) Hz. Based on this value, it takes about 7.8 cycles for the shedding to start.

Using the peak frequency value of \(f = 0.0888\) Hz, \(St\) was evaluated. The table below compares the computed value from SLIM with that of the experiment. The agreement is good given that experimental uncertainty can be relatively high at low Reynolds numbers.

Frequency (Hz) | Strouhal Number | |
---|---|---|

Experimental | 0.0835 | \(0.167\) |

SLIM | 0.0888 | \(0.177\) |

**Conclusions**

The transiet, incompressible, two-dimensional flow over a circular cylinder was simulated using Caelus 4.10 to estimate the peak frequency of vortex shedding. The value was compared to well known experimental data resulting in good agreement.

*Laminar, incompressible flow inside a lid driven Triangular Cavity*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(D\) | Depth of the cavity | \(m\) |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_D\) | Reynolds number based on depth | Non-dimensional |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(W\) | Width | \(m\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

This validation study concerns the laminar, incompressible flow inside a lid driven triangular cavity. Here, the top wall of the cavity moves at a constant velocity initiating a recirculating motion within the cavity. Due to the viscous nature of the flow, a boundary layer develops in the direction of the moving lid, while flow reversal occurs due to the recirculating flow. The flow feature of interest is the velocity distribution along the centre-line of the cavity.

Benchmark experiments on this configuration has been reported in Jyotsna and Venka [11] for a Reynolds number of 800. The main objective of this validation case was to compare the \(x\) velocity distribution against experimental data.

**Problem Definition**

A schematic of the triangular cavity is presented in Fig. 12 where depth of the cavity D = 4 m and the width W = 2 m. The Reynolds number based on the cavity depth is 800 and the wall velocity is U = 2 m/s. Using the Reynolds number, U, and D, kinematic viscosity was calculated to be 0.01 \(m^2/s\).

The table below summaries the flow properties

\(Re_D\) | \(D~(m)\) | \(W~(m)\) | \(U~(m/s)\) | \(p~(m^2/s^2)\) | \(\nu~(m^2/s)\) |
---|---|---|---|---|---|

800 | 4.0 | 2.0 | 2.0 | \((0)\) Gauge | \(0.01\) |

The flow in this case is assumed to be incompressible and hence the density remained constant throughout. Further, the temperature field is not accounted into the calculation, and therefore the viscosity of the flow can also remain constant. Since viscosity is constant, it becomes more convenient to specify it as kinematic viscosity. It should be noted that in Caelus for incompressible flows, both pressure and viscosity are always specified as kinematic.

**Computational Domain and Boundary Conditions**

The computational domain is the triangular cavity shown in Fig. 13. Highlighted in blue, the side walls of the cavity have a no-slip boundary condition while the top wall, highlighted in green, has a uniform velocity in the \(x\) direction.

*Boundary Conditions and Initialisation*

- Moving wall
- Velocity: Fixed uniform velocity \(u = 2.0~m/s\) in \(x\) direction
- Pressure: Zero gradient

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient

- Initialisation
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero Gauge pressure

**Computational Grid**

The 2D grid in \(x-y\) plane is shown in Fig. 14. A hybrid grid is employed for this case with a total of 5538 cells. Up to a depth of D = 1.35 m a structured grid is used while below that value an unstructured triangular grid is used. An unstructured grid is used in the bottom portion because it resulted in lower skewness in this vicinity. For the structured region, 40 cells are distributed across the width of the cavity and 40 along the depth. The cavity walls in the unstructured region have 100 cells along each. The interface of the two regions is *node matched* and has 40 cells across the width. The grid close to the cavity lid was refined to better capture the shear layer.

The flow characteristics in the cavity can be assumed to be two dimensional and here it has been solved with the same assumption. Since Caelus is a 3D computational framework, it necessitates the grid to also be 3D. Therefore, a 3D grid was obtained using Pointwise by extruding the 2D grid in the positive \(z\) direction by *one cell*. The final 3D grid was then exported to the Caelus format (polyMesh). The two \(x-y\) planes obtained as a result of grid extrusion need boundary conditions to be specified. As the flow over a flat-plate is generally 2D, we do not need to solve the flow in the third dimension. This is achieved in Caelus by specifying *empty* boundary condition for each plane. Although, no flow is computed in the \(z\) direction, a velocity of \(w = 0\) has to be specified for the velocity boundary condition as indicated above.

**Results and Discussion**

A steady solution to the cavity was obtained using Caelus 4.10 with the SLIM solver. While a time-dependent approach was used, the solution was simulated sufficiently long so that steady flow was achieved. To determine when this occured the velocity distribution along the cavity centre-line was monitored with respect to time. The simulations was stopped when no appreciable changes were observed.

In Fig. 15, the \(x\) velocity distribution along the cavity centre-line is compared with that of the benchmark experimental data. The \(y\) distance is normalised with the cavity depth (\(D\)) which gives \(y/d = 0\) at the cavity lid and \(y/d = -1\) at the bottom vertex. Similarly, the \(u\) velocity is normalised with the velocity of the cavity lid (\(u_L\)).

As seen in Fig. 15 above, the comparison the experiment is excellent.

**Conclusions**

A steady, incompressible flow past a two-dimensional triangular cavity was simulated using Caelus 4.10 on a hybrid grid. The velocity distribution along the centre-line of the cavity was compared with the benchmark experimental data, it was found that the SLIM results compared very favorably.

*Natural Convection in a Spherical Cavity*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(C_p\) | Specific heat at constant pressure | \(J/kg \cdot K\) |

\(D\) | Diameter of the sphere | \(m\) |

\(g\) | Gravitational constant | \(9.80~m/s^2\) |

\(k\) | Thermal conductivity | \(W/m \cdot K\) |

\(Nu\) | Nusselt number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Pr\) | Prandtl number | Non-dimensional |

\(r\) | Radius of the sphere | \(m\) |

\(Ra\) | Rayleigh number | Non-dimensional |

\(T\) | Temperature | \(K\) |

\(u\) | velocity in x-direction | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(z\) | Distance in z-direction | \(m\) |

\(\beta\) | Coefficient of thermal expansion | \(1/K\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

In this study, laminar flow inside a heated spherical cavity is investigated. A temperature gradient was applied to the surface of the cavity inducing fluid motion due to buoyancy effects. Here, a Rayleigh number (\(Ra\)) of 2000 and a Prandtl number (\(Pr\)) of 0.7 is used to set appropriate thermal conditions to the problem.

Analytical solutions to the heated spherical cavity were investigated by McBain and Stephens [4] at various Rayleigh numbers up to \(Ra = 10000\). However, it was noted that asymptotic heat transfer rate predictions as a function of Nusselt number (\(Nu\)) deviated largely with increase in Rayleigh number. Therefore a value of \(Ra = 2000\) was chosen to avoid these deviations. At this value, the analytical heat transfer rate compares very well with the asymptotic solution. Isotherms from the analytical solution were used for comparision with numerical results.

**Problem Definition**

The schematic of the spherical cavity is depicted Fig. 16 and as can be seen, only half of the sphere is considered here with an \(x-y\) symmetry plane. The sphere is located at \(x = 0, y = 0, z = 0\) with a radius \(r = 0.5~m\).

As indicated, a non-uniform temperature profile was used as a thermal boundary condition on the spherical wall. The temperature (\(T\)) was specified as a function of \(x\):

\[T = x\]

with,

\[T = -0.5~K \quad \text{at} \quad x = -0.5~m\]

\[T = 0.5~K \quad \text{at} \quad x = 0.5~m\]

The energy equation in a non-dimensional form [4] for a steady state can be expressed as:

\[Gr~Pr~u \cdot T = \nabla^2 T\]

where, \(Gr\) is the Grashof number. Since, \(Ra = Gr~Pr\) the above equation can be rewritten along with expanding the Laplacian term (\(\nabla^2\)) as

\[Ra~u \cdot T = \nabla~[\alpha \nabla T]\]\[\text{or}\]\[u \cdot T = \frac{1}{Ra}~\alpha~\nabla~[ \nabla T]\]

where \(\alpha\) is the thermometric conductivity. It is reasonable to assume that (\(1/Ra\)) is approximately equal to the thermometric conductivity (\(\alpha\)), which is given as

\[\alpha = \frac{k}{\rho C_p} = \frac{\nu}{Pr}\]

where, \(k\), \(\rho\), \(C_p\) and \(\nu\) are the thermal conductivity, density, specific heat capacity and kinematic viscosity of the fluid respectively. Using the above relation with a value of \(Ra = 2000\) and \(Pr = 0.7\), the kinematic viscosity was calculated to be \(\nu = 3.4 \times 10^{-4}~m^2/s\). The coefficient of thermal expansion (\(\beta\)) needed to model the Boussinesq buoyancy term was evaluated from the following relation

\[Gr = \frac{g~\beta~\Delta T~D^3}{\nu^2}\]

where, \(g\) is the acceleration due to gravity and \(D\) is the diameter of the sphere. \(\beta\) was calculated to \(3.567\times10^{-5}~1/K\). In the following table, a summary of the properties are given. Note that gravity acts in \(-y\) direction.

\(Ra\) | \(Pr\) | \(T~(K)\) | \(p~(m^2/s^2)\) | \(\nu~(m^2/s)\) | \(\beta~(1/K)\) |
---|---|---|---|---|---|

\(2000\) | \(0.7\) | \(T = x\) | \((0)\) Gauge | \(3.4 \times 10^{-4}\) | \(3.567\times10^{-5}\) |

Although the temperature is calculated in this simulation, a constant viscosity is used. Since the temperature gradient is very small (\(\mathcal{O}(1)\)), effect of temperature on the viscosity would be insignificant. The kinematic definition of pressure is used here.

**Computational Domain and Boundary Conditions**

The computational domain was a half sphere with an \(x-y\) plane of symmetry at \(z = 0~m\). The surface temperature was prescribed as discussed above. The initialisation of the fluid temperature within the sphere follows that of the surface temperature (\(T = x\)) and is depicted in Fig. 17 at the symmetry plane. Note that this figure also aids in providing a clarity of understanding for the temperature variation over the spherical surface.

*Boundary Conditions and Initialisation*

- Wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Uniform zero Buoyant Pressure
- Temperature: Linear unction of \(x\) (\(T = x\))

- Symmetry Plane
- Velocity: Symmetry
- Pressure: Symmetry
- Temperature: Symmetry

- Initialisation
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Uniform zero Buoyant Pressure
- Temperature: Linear function of \(x\) (\(T = x\))

**Computational Grid**

The computational grid for the half sphere was generated using Pointwise. A fully structured grid was constructured with a total of 18564 cells. As seen in Fig. 18, an O-H topology used where an H-block is centred within 5 O-blocks.

**Results and Discussion**

The steady solution to the natural convection in a buoyant sphere was obtained using Caelus 4.10 with the SLIM solver that includes the a buoyancy source term based on the Boussinesq assumption. Since SLIM is inherently time-accurate, the simulation was run sufficiently long such that a steady state was achieved. In Fig. 19, compares the isotherms obtained with SLIM and the analytical isotherms obtained with a first order approximation. Close agreement was observed.

**Conclusions**

A validation study of a buoyant flow inside a spherically heated cavity was conducted using Caelus 4.10. The isotherms obtained from the CFD results were compared with the first order analytical solution and excellent agreement was observed.

This section discusses the turbulence model verification and validation conducted for Caelus 4.10. The implemented Spalart–Allmaras, Spalart–Allmaras Curvature Correction, \(k-\omega~\rm{SST}\) turbulence models are verified with NASA’s CFL3D results. To validate these models, the Caelus results are compared with available experimental data. The cases considered for this exercise are obtained from the Turbulence Modeling Resource database.

Turbulent, incompressible flow over a two-dimensional Sharp-Leading Edge flat-plate

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(a\) | Speed of sound | \(m/s\) |

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(k\) | Turbulent kinetic energy | \(m^2/s^2\) |

\(L\) | Length of the plate | \(m\) |

\(M_\infty\) | Freestream Mach number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_L\) | Reynolds number | \(1/m\) |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(y^+\) | Wall distance | Non-dimensional |

\(\mu\) | Dynamic viscosity | \(kg/m~s\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\tilde{\nu}\) | Turbulence field variable | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

\(\omega\) | Specific dissipation rate | \(1/s\) |

\(\infty\) | Freestream conditions | |

\(t\) (subscript) | Turbulent property |

**Introduction**

In this case, steady turbulent incompressible flow over a two-dimensional sharp-leading edge flat-plate is considered at zero angle of incidence, which generates a turbulent boundary layer with zero-pressure gradient over the flat plate. The simulations are carried over a series of four successively refined grids and the solutions are compared with the CFL3D data. This therefore serves as both grid independence study and verification of the turbulence models. The distribution of skin-friction coefficient (\(c_f\)) along the plate is used to verify the accuracy of the models.

**Problem definition**

This exercise is based on the Turbulence Modeling Resource case for a flat-plate and follows the same conditions used in the incompressible limit. However, note that CFL3D uses a freestream Mach number (\(M_\infty\)) of 0.2 as it is a compressible solver. The schematic of the geometric configuration is shown in Fig. 20.

The length of the plate is L = 2.0 m, wherein, x = 0 is at the leading edge and the Reynolds number is \(5 \times 10^6\) and \(U_\infty\) is the freestream velocity in the x -direction. The inflow temperature in this case is assumed to be 300 K and for Air as a perfect gas, the speed of sound (\(a\)) can be evaluated to 347.180 m/s. Based on the freestream Mach number and speed of sound, velocity can be calculated, which is \(U_\infty\) = 69.436 m/s. Using velocity and Reynolds number, the kinematic viscosity can be evaluated. The following table summarises the freestream conditions used:

Fluid | \(Re_L~(1/m)\) | \(U_\infty~(m/s)\) | \(p_\infty~(m^2/s^2)\) | \(T_\infty~(K)\) | \(\nu_\infty~(m^2/s)\) |
---|---|---|---|---|---|

Air | \(5 \times 10^6\) | 69.436113 | \((0)\) Gauge | 300 | \(1.38872\times10^{-5}\) |

Note that in an incompressible flow the density (\(\rho\)) does not change and is reasonable to assume a constant density throughout the calculation. In addition, temperature is not considered and therefore the viscosity can also be held constant. In Caelus, for incompressible flows, pressure and viscosity are always specified as kinematic.

*Turbulent Properties for Spalart–Allmaras model*

The turbulent inflow boundary conditions used for the Spalart–Allmaras model were calculated as \(\tilde{\nu}_{\infty} = 3 \cdot \nu_\infty\) and subsequently turbulent eddy viscosity was evaluated. The following table provides the values of these used in the current simulations:

\(\tilde{\nu}_\infty~(m^2/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|

\(4.166166 \times 10^{-5}\) | \(2.9224023 \times 10^{-6}\) |

*Turbulent Properties for k-omega SST model*

The turbulent inflow boundary conditions used for \(k-\omega~\rm{SST}\) were calculated as follows and is as given in Turbulence Modeling Resource

\[k_{\infty} = 9 \times 10^{-9} \cdot a^2_\infty = \frac{1.125 U_\infty^2}{Re_L}\]

\[\omega_{\infty} = 1 \times 10^{-6} \cdot \frac{\rho_\infty a^2_\infty}{\mu_\infty} = \frac{125 U_\infty}{L}\]

\[\nu_{t~\infty} = 0.009 \times \nu_\infty\]

Note that the dynamic viscosity for the above equation is obtained from Sutherland formulation and density is calculated as \(\rho = \mu / \nu\). The below table provides the turbulent properties used in the current simulations

\(k_{\infty}~(m^2/s^2)\) | \(\omega_{\infty}~(1/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|---|

\(1.0848 \times 10^{-3}\) | \(8679.5135\) | \(1.24985 \times 10^{-7}\) |

**Computational Domain and Boundary Conditions**

The computational domain is a rectangular block encompassing the flat-plate. Fig. 21 below shows the details of the boundaries used in two-dimensions (\(x-y\) plane). As can be seen, the region of interest (highlighted in blue) extends between \(0\leq x \leq 2.0~m\) and has a no-slip boundary condition. Upstream of the leading edge, a symmetry boundary is used to simulate a freestream flow approaching the flat-plate. The inlet boundary as shown in Fig. 21 is placed at start of the symmetry at \(x = -0.3333~m\) and the outlet at the exit plane of the no-slip wall (blue region) at \(x = 2.0~m\). A symmetry plane condition is used for the entire top boundary.

*Boundary Conditions and Initialisation*

Following are the boundary condition details used for the computational domain:

- Inlet
- Velocity: Fixed uniform velocity \(u = 69.436113~m/s\) in \(x\) direction
- Pressure: Zero gradient
- Turbulence:
- Spalart-Allmaras (Fixed uniform values of \(\nu_{t~\infty}\) and \(\tilde{\nu}_{\infty}\) as given in the above table)
- \(k-\omega~\textrm{SST}\) (Fixed uniform values of \(k_{\infty}\), \(\omega_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

- Symmetry
- Velocity: Symmetry
- Pressure: Symmetry
- Turbulence: Symmetry

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient
- Turbulence:
- Spalart-Allmaras (Fixed uniform values of \(\nu_{t}=0\) and \(\tilde{\nu}=0\))
- \(k-\omega~\textrm{SST}\) (Fixed uniform values of \(k = 0\) and \(\nu_t=0\); \(\omega\) = omegaWallFunction)

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)
- Turbulence:
- Spalart-Allmaras (Calculated \(\nu_{t}=0\) and Zero gradient \(\tilde{\nu}\))
- \(k-\omega~\textrm{SST}\) (Zero gradient \(k\) and \(\omega\); Calculated \(\nu_t=0\); )

- Initialisation
- Velocity: Fixed uniform velocity \(u = 69.436113~m/s\) in \(x\) direction
- Pressure: Zero Gauge pressure
- Turbulence:
- Spalart-Allmaras (Fixed uniform values of \(\nu_{t~\infty}\) and \(\tilde{\nu}_{\infty}\) as given in the above table)
- \(k-\omega~\textrm{SST}\) (Fixed uniform values of \(k_{\infty}\), \(\omega_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

**Computational Grid**

The 3D structured grid was obtained from Turbulence Modeling Resource as a Plot3D and was converted to Caelus format using Pointwise. It should be noted that the flow normal direction in the Plot3D grids is \(z\) and the two-dimensional plane of interest is in \(x-z\) directions. Since the flow-field of interest is two-dimensional, and simpleSolver being a 3D solver, the two \(x-z\) planes are specified with empty boundary conditions. As mentioned earlier, a series of four grids were considered from the original set of five, excluding the coarsest. Details of the different grids used are given below.

Grid | Cells in \(x\)-direction | Cells in \(z\)-direction | Total | \(y^+\) |
---|---|---|---|---|

Grid-2 | 68 | 48 | 3264 | 0.405 |

Grid-3 | 136 | 96 | 13,056 | 0.203 |

Grid-4 | 272 | 192 | 52,224 | 0.101 |

Grid-5 | 544 | 384 | 208,896 | 0.05 |

In Fig. 22, the 2D grid in the \(x-z\) plane is shown for Grid-4. As can be seen, the grid is refined close to the wall in order to capture the turbulent boundary layer accurately. All grids have \(y^+ < 1\) and no wall function is used for the wall boundary in the current verification cases.

**Results and Discussion**

The steady-state solution of the turbulent flow over a flat plate was obtained using Caelus 4.10. The simpleSolver was used here and the solution was simulated for a sufficient length until the residuals for pressure, velocity and turbulent quantities were less than \(1 \times 10^{-6}\). The finite volume discretization of the gradient of pressure and velocity was carried out using the linear approach. Where as the divergence of velocity and mass flux was carried out through the linear upwind method. However, for the divergence of the turbulent quantities, upwind approach was utilised and linear approach for the divergence of the Reynolds stress terms. For the discretization of the Laplacian terms, again linear corrected method was used. For some grids having greater than 50 degree non-orthogonal angle, linear limited with a value of 0.5 was used for the Laplacian of the turbulent stress terms.

*Spalart–Allmaras*

In Fig. 23, the skin-friction distribution along the flat-plate obtained from Caelus for different grids is shown. As can be seen, all grids produce the same skin-friction values suggesting a grid-independent solution is achieved.

In Fig. 24, the skin-friction distribution obtained from Caelus on Grid-5 is compared with CFL3D of the same grid. An excellent agreement is obtained all along the plate.

*k-Omega SST*

The skin-friction distribution for various grids obtained from \(k-\omega~\rm{SST}\) model is shown in Fig. 25.

The skin-friction comparison between Caelus and CFL3D for Grid-5 is shown in Fig. 26.

**Conclusions**

The steady turbulent flow over a two-dimensional flat-plate was simulated using Caelus 4.10 utilising the simpleSolver. The simulations were carried out with two turbulence models and the obtained solutions were verified against CFL3D data. The results were found to be in good agreement with CFL3D and suggesting the turbulence implementation in Caelus is accurate.

*Turbulent, incompressible flow over a two-dimensional bump in a channel*

Symbol | Definition | Units (SI) |
---|---|---|

\(a\) | Speed of sound | \(m/s\) |

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(k\) | Turbulent kinetic energy | \(m^2/s^2\) |

\(L\) | Length of the bump | \(m\) |

\(M_\infty\) | Freestream Mach number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_L\) | Reynolds number | \(1/m\) |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(z\) | Distance in z-direction | \(m\) |

\(y^+\) | Wall distance | Non-dimensional |

\(\mu\) | Dynamic viscosity | \(kg/m~s\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\tilde{\nu}\) | Turbulence field variable | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

\(\omega\) | Specific dissipation rate | \(1/s\) |

\(\infty\) | Freestream conditions | |

\(t\) (subscript) | Turbulent property |

**Introduction**

This case covers the verification of turbulent incompressible flow over a two-dimensional bump in a channel. The bump acts as a perturbation causing local changes to the velocity and pressure over the bump surface. Since the perturbation is quite small, the flow remains attached to the bump surface. The simulations are carried out over a series of four grids which are successively refined and are studied over two turbulence models, serving as a grid independence study. The distribution of skin-friction (\(c_f\)) is then compared with that of the CFL3D data.

**Problem definition**

This verification exercise is based on the Turbulence Modeling Resource case for a 2D bump in a channel and follows the same conditions used in the limit of incompressibility. In this case, CFL3D uses a freestream Mach number (\(M_\infty\)) of 0.2. The schematic of the geometric configuration is shown in Fig. 27.

The location of the plate upstream of the bump begins at x=0 m and ends at x = 1.5 m, giving a total plate length of L = 1.5 m. The flow has a Reynolds number of \(3 \times 10^6\) with a freestream velocity \(U_\infty\) in the x-direction. The temperature of the inflow is assumed to be 300 K and for Air as a perfect gas, the speed of sound (\(a\)) can be evaluated to 347.180 m/s. With the freestream Mach number and speed of sound, the freestream velocity was calculated to \(U_\infty\) = 69.436 m/s. Kinematic viscosity was then obtained from velocity and Reynolds number. The following table summarises the freestream conditions used for this case.

Fluid | \(Re_L~(1/m)\) | \(U_\infty~(m/s)\) | \(p_\infty~(m^2/s^2)\) | \(T_\infty~(K)\) | \(\nu_\infty~(m^2/s)\) |
---|---|---|---|---|---|

Air | \(3 \times 10^6\) | 69.436113 | \((0)\) Gauge | 300 | \(2.314537\times10^{-5}\) |

Since the flow is incompressible, the density (\(\rho\)) does not change and therefore, a constant density is assumed throughout the calculation. Further, the temperature is not considered and hence it does not have any influence on viscosity, which is therefore kept constant. Note that in Caelus, pressure and viscosity are always specified as kinematic for a incompressible flow simulation.

*Turbulent Properties for Spalart–Allmaras model*

The turbulent inflow boundary conditions used for the Spalart–Allmaras model were calculated as \(\tilde{\nu}_{\infty} = 3 \cdot \nu_\infty\) and subsequently turbulent eddy viscosity was evaluated. The following table provides the values of these used in the current simulations:

\(\tilde{\nu}_\infty~(m^2/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|

\(6.943611 \times 10^{-5}\) | \(4.8706713 \times 10^{-6}\) |

*Turbulent Properties for k-omega SST model*

The turbulent inflow boundary conditions used for \(k-\omega~\rm{SST}\) were calculated as follows and is as given in Turbulence Modeling Resource

\[k_{\infty} = 9 \times 10^{-9} \cdot a^2_\infty = \frac{0.675 U_\infty}{Re_L}\]

\[\omega_{\infty} = 1 \times 10^{-6} \cdot \frac{\rho_\infty a^2_\infty}{\mu_\infty} = \frac{50 U_\infty}{L}\]

\[\nu_{t~\infty} = 0.009 \times \nu_\infty\]

Note that the dynamic viscosity for the above equation is obtained from Sutherland formulation and density is calculated as \(\rho = \mu / \nu\). The below table provides the turbulent properties used in the current simulations:

\(k_{\infty}~(m^2/s^2)\) | \(\omega_{\infty}~(1/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|---|

\(1.0848 \times 10^{-3}\) | \(5207.6475\) | \(2.08310 \times 10^{-7}\) |

**Computational Domain and Boundary Conditions**

The computational domain consists of a rectangular channel encompassing the bump. In Fig. 28 , the details of the boundaries used in two-dimensions (\(x-y\) plane) are shown. The region of interest, which is the bump extends between \(0\leq x \leq 1.5~m\) and has a no-slip boundary condition. Upstream and downstream of the bump, the symmetry boundary extends about 17 bump lengths. The inlet boundary is placed at the start of the symmetry at \(x = -25.0~m\) and the outlet is placed at \(x = 26.5~m\). For the entire top boundary, symmetry plane condition is used.

*Boundary Conditions and Initialisation*

Following are the boundary condition details used for the computational domain:

- Inlet
- Velocity: Fixed uniform velocity \(u = 69.436113~m/s\) in \(x\) direction
- Pressure: Zero gradient
- Turbulence:
- Spalart–Allmaras (Fixed uniform values of \(\nu_{t~\infty}\) and \(\tilde{\nu}_{\infty}\) as given in the above table)
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k_{\infty}\), \(\omega_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

- Symmetry
- Velocity: Symmetry
- Pressure: Symmetry
- Turbulence: Symmetry

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient
- Turbulence:
- Spalart–Allmaras (Fixed uniform values of \(\nu_{t}=0\) and \(\tilde{\nu}=0\))
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k = 0\) and \(\nu_t=0\); \(\omega\) = omegaWallFunction)

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)
- Turbulence:
- Spalart–Allmaras (Calculated \(\nu_{t}=0\) and Zero gradient \(\tilde{\nu}\))
- \(k-\omega~\rm{SST}\) (Zero gradient \(k\) and \(\omega\); Calculated \(\nu_t=0\); )

- Initialisation
- Velocity: Fixed uniform velocity \(u = 69.436113~m/s\) in \(x\) direction
- Pressure: Zero Gauge pressure
- Turbulence:
- Spalart–Allmaras (Fixed uniform values of \(\nu_{t~\infty}\) and \(\tilde{\nu}_{\infty}\) as given in the above table)
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k_{\infty}\), \(\omega_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

**Computational Grid**

The 3D computational grid was obtained from Turbulence Modeling Resource as a Plot3D and was converted to Caelus format using Pointwise. In the Plot3D computational grid, the flow normal direction is \(z\) and thus the two-dimensional plane of interest is in \(x-z\) directions. Further, since the flow-field is of two-dimensional, and the simpleSolver being a 3D solver, the two \(x-z\) planes are specified with empty boundary conditions. A series of four grids were considered from the original set of five, excluding the coarsest grid and the following table give its details.

Grid | Cells in \(x\)-direction | Cells in \(z\)-direction | Total | \(y^+\) |
---|---|---|---|---|

Grid-2 | 176 | 80 | 14,080 | 0.236 |

Grid-3 | 352 | 160 | 56,320 | 0.118 |

Grid-4 | 704 | 320 | 225,280 | 0.059 |

Grid-5 | 1408 | 640 | 901,120 | 0.03 |

The 2D grid in \(x-z\) plane is shown in Fig. 29 for Grid-3. As can be noted, the grid is sufficiently refined close to the wall in the normal direction. In addition, the grids are refined in the vicinity of the bump, including both upstream and downstream which can be seen in the inset. All grids have a \(y^+ < 1\) and no wall function is used for the wall boundary in the current verification cases.

**Results and Discussion**

The steady-state solution of the turbulent flow over a two-dimensional bump was obtained using Caelus 4.10. The simpleSolver was used for the calculations and was run for a sufficient length until the residuals for pressure, velocity and turbulent quantities were less than \(1 \times 10^{-6}\). The finite volume discretization of the gradient of pressure and velocity was carried out using the linear approach. Where as the divergence of velocity and mass flux was carried out through the linear upwind method. However, for the divergence of the turbulent quantities, upwind approach was utilised and linear approach for the divergence of the Reynolds stress terms. For the discretization of the Laplacian terms, again linear corrected method was used. For some grids having greater than 50 degree non-orthogonal angle, linear limited with a value of 0.5 was used for the Laplacian of the turbulent stress terms.

*Spalart–Allmaras*

The skin-friction distributions over the 2D bump obtained from Caelus for different grids are shown in Fig. 30. There is very little difference in the skin-friction beyond Grid-2 suggesting that a grid-independence solution is achieved.

In Fig. 31 , the comparison between Caelus and CFL3D is made for Grid-5 and as can be seen, a very good agreement is obtained over the entire region of the bump.

*k-Omega SST*

The skin-friction distribution variation for different grids obtained from \(k-\omega~\rm{SST}\) model is shown in Fig. 32.

In Fig. 33 , the skin-friction comparison between Caelus and CFL3D is made for Grid-5 and is shown.

**Conclusions**

The steady turbulent flow simulation over a two-dimensional bump was carried out using Caelus 4.10 employing simpleSolver. The solutions were obtained with two turbulence models, implemented in-house and the results were verified against CFL3D data. The comparison was found to be in good agreement with CFL3D suggesting that the turbulence model implementation is accurate in Caelus.

**Nomenclature**

*Turbulent, incompressible flow over a two-dimensional NACA airfoil*

Symbol | Definition | Units (SI) |
---|---|---|

\(a\) | Speed of sound | \(m/s\) |

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(c_p\) | Pressure coefficient | Non-dimensional |

\(C\) | Chord length | \(m\) |

\(I\) | Turbulent intensity | Percentage |

\(k\) | Turbulent kinetic energy | \(m^2/s^2\) |

\(M_\infty\) | Freestream Mach number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_L\) | Reynolds number | \(1/m\) |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(w\) | Local velocity in z-direction | \(m/s\) |

\(U\) | Freestream velocity | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(z\) | Distance in y-direction | \(m\) |

\(y^+\) | Wall distance | Non-dimensional |

\(\alpha\) | Angle of attack | Degrees |

\(\mu\) | Dynamic viscosity | \(kg/m~s\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\tilde{\nu}\) | Turbulence field variable | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

\(\omega\) | Specific dissipation rate | \(1/s\) |

\(\infty\) | Freestream conditions | |

\(t\) (subscript) | Turbulent property |

**Introduction**

This case deals with the steady turbulent incompressible flow over a two-dimensional NACA 0012 airfoil. The study is conducted at two angles of attack, \(\alpha = 0^\circ\) and \(\alpha = 10^\circ\) respectively. The simulations are carried out over a series of four grids and compared with CFL3D data and the results are also compared with the experimental data. This exercise therefore verifies and validates the turbulence models used through the distribution of skin-friction coefficient (\(c_f\)) and pressure coefficient (\(c_p\)) over the airfoil surface.

**Problem definition**

This verification and validation exercise is based on the Turbulence Modeling Resource case for a NACA 0012 airfoil and follows the same flow conditions used in the incompressible limit. However, the numerical code CFL3D uses a freestream Mach number (\(M_\infty\)) of 0.15. In Fig. 34 the schematic of the airfoil is shown. Note that the 2D plane in Fig. 34 is depicted in \(x-z\) directions as the computational grid also follows the same 2D plane.

The length of the airfoil chord is C = 1.0 m, wherein, x = 0 is at the leading edge and the Reynolds number is \(6 \times 10^6\) and \(U_\infty\) is the freestream velocity. For \(\alpha = 10^\circ\), the velocity components are evaluated in order to have a same resultant freestream velocity. The freestream flow temperature in this case is assumed to be 300 K and for Air as a perfect gas, the speed of sound (\(a\)) can be evaluated to 347.180 m/s. Based on the freestream Mach number and speed of sound, freestream velocity can be evaluated to \(U_\infty\) = 52.077 m/s. Using the value of velocity and Reynolds number, kinematic viscosity can be calculated. The following table summarises the freestream conditions used:

Fluid | \(Re_L~(1/m)\) | \(U_\infty~(m/s)\) | \(p_\infty~(m^2/s^2)\) | \(T_\infty~(K)\) | \(\nu_\infty~(m^2/s)\) |
---|---|---|---|---|---|

Air | \(6 \times 10^6\) | 52.0770 | \((0)\) Gauge | 300 | \(8.679514\times10^{-6}\) |

To get a freestream velocity of \(U_\infty\) = 52.077 m/s at \(\alpha = 10^\circ\), the velocity components in \(x\) and \(z\) are resolved. These are provided in the table below

\(\alpha~\rm{(Degrees)}\) | \(u~(m/s)\) | \(w~(m/s)\) |
---|---|---|

\(0^\circ\) | 52.0770 | 0.0 |

\(10^\circ\) | 51.2858 | 9.04307 |

Note that in an incompressible flow, the density (\(\rho\)) does not vary and a constant density can be assumed throughout the calculation. Further, since temperature is not considered here, the viscosity is also held constant. In Caelus for incompressible flow simulations, pressure and viscosity are always specified as kinematic.

*Turbulent Properties for Spalart–Allmaras model*

The turbulent inflow boundary conditions used for the Spalart–Allmaras model were calculated as \(\tilde{\nu}_{\infty} = 3 \cdot \nu_\infty\) and subsequently turbulent eddy viscosity was evaluated. The following table provides the values of these used in the current simulations:

\(\tilde{\nu}_\infty~(m^2/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|

\(2.603854 \times 10^{-5}\) | \(1.8265016 \times 10^{-6}\) |

*Turbulent Properties for k-omega SST model*

The turbulent inflow boundary conditions used for \(k-\omega~\rm{SST}\) were calculated as follows and is as given in Turbulence Modeling Resource

\[k_{\infty} = \frac{3}{2} (U_\infty I)^2\]

\[\omega_{\infty} = 1 \times 10^{-6} \cdot \frac{\rho_\infty a^2_\infty}{\mu_\infty} = \frac{266.7 U_\infty}{L}\]

\[\nu_{t~\infty} = 0.009 \times \nu_\infty\]

Note that the dynamic viscosity in the above equation is obtained from Sutherland formulation and density is evaluated as \(\rho = \mu / \nu\). In the below table, the turbulent properties used in the current simulations are provided.

\(I\) | \(k_{\infty}~(m^2/s^2)\) | \(\omega_{\infty}~(1/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|---|---|

\(0.052\%\) | \(1.0999 \times 10^{-3}\) | \(13887.219\) | \(7.811564 \times 10^{-8}\) |

**Computational Domain and Boundary Conditions**

The computational domain used for the airfoil simulations and the details of the boundaries are shown in Fig. 35 for a \(x-z\) plane. The leading edge and the trailing edge extends between \(0 \leq x \leq 1.0~m\) and the entire airfoil has a no-slip boundary condition. The far-field domain extends by about 500 chord lengths in the radial direction and the inlet is placed for the entire boundary highlighted in green. The outlet boundary is placed at the exit plane, which is at \(x \approx 500~m\).

*Boundary Conditions and Initialisation*

- Inlet
- Velocity:
- \(\alpha=0^\circ\): Fixed uniform velocity \(u = 52.0770~m/s\); \(v = w = 0.0~m/s\) in \(x, y\) and \(z\) directions respectively
- \(\alpha=10^\circ\): Fixed uniform velocity \(u = 51.2858~m/s\); \(v = 0.0~m/s\) and \(w = 9.04307~m/s\) in \(x, y\) and \(z\) directions respectively
- Pressure: Zero gradient
- Turbulence:
- Spalart–Allmaras (Fixed uniform values of \(\nu_{t~\infty}\) and \(\tilde{\nu}_{\infty}\) as given in the above table)
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k_{\infty}\), \(\omega_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

- Velocity:

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient
- Turbulence:
- Spalart–Allmaras (Fixed uniform values of \(\nu_{t}=0\) and \(\tilde{\nu}=0\))
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k = <<0\) and \(\nu_t=0\); \(\omega\) = omegaWallFunction)

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)
- Turbulence:
- Spalart–Allmaras (Calculated \(\nu_{t}=0\) and Zero gradient \(\tilde{\nu}\))
- \(k-\omega~\rm{SST}\) (Zero gradient \(k\) and \(\omega\); Calculated \(\nu_t=0\); )

- Initialisation
- Velocity:
- \(\alpha=0^\circ\): Fixed uniform velocity \(u = 52.0770~m/s\); \(v = w = 0.0~m/s\) in \(x, y\) and \(z\) directions respectively
- \(\alpha=10^\circ\): Fixed uniform velocity \(u = 51.2858~m/s\); \(v = 0.0~m/s\) and \(w = 9.04307~m/s\) in \(x, y\) and \(z\) directions respectively
- Pressure: Zero Gauge pressure
- Turbulence:

- Velocity:

**Computational Grid**

The 3D computational grid for the NACA 0012 airfoil was obtained from Turbulence Modeling Resource as a Plot3D format. Using Pointwise it was then converted to Caelus format. As indicated earlier, the two-dimensional plane of interest in the Plot3D grid is in \(x-z\) directions. As the flow is considered here to be two-dimensional, and simpleSolver being a 3D solver, the two \(x-z\) planes are specified with empty boundary conditions consequently treating as symmetry flow in \(y\) direction. To study the sensitivity of the grid, four grids were considered from the original set of five, in which the coarsest grid was excluded from this study. Details of the different grids used are given in the below table. Not that for both angles of attack, same grid is used.

Grid | Cells over airfoil | Cells in normal direction | Total | \(y^+\) |
---|---|---|---|---|

Grid-2 | 128 | 64 | 14,336 | 0.465 |

Grid-3 | 256 | 128 | 57,344 | 0.209 |

Grid-4 | 512 | 256 | 229,376 | 0.098 |

Grid-5 | 1024 | 512 | 917,504 | 0.047 |

The below Fig. 36 shows the 2D grid in \(x-z\) plane for Grid-3 and the refinement around the airfoil is shown in the inset. Sufficient refinement can be seen in the wall normal direction and all the grid have a \(y^+ < 1\) and no wall function is used for the airfoil surface throughout the current verification and validation cases.

**Results and Discussion**

The solution to the turbulent flow over the NACA 0012 airfoil was obtained using Caelus 4.10. SimpleSolver was used and the solutions were run sufficiently long until the residuals for pressure, velocity and turbulence quantities were less than \(1 \times 10^{-6}\). The finite volume discretization of the gradient of pressure and velocity was carried out using the linear approach. Where as the divergence of velocity and mass flux was carried out through the linear upwind method. However, for the divergence of the turbulent quantities, upwind approach was utilised and linear approach for the divergence of the Reynolds stress terms. For the discretization of the Laplacian terms, again linear corrected method was used. For some grids having greater than 50 degree non-orthogonal angle, linear limited with a value of 0.5 was used for the Laplacian of the turbulent stress terms.

The verification results are shown firstly for both angles of attack and is followed by the experimental validation data.

*Verification results: Spalart–Allmaras*

The following Fig. 37 and Fig. 38 shows the skin-friction distribution over the upper surface for \(\alpha=0^\circ\) and \(\alpha=10^\circ\) from Caelus for different grids. In both cases, Grid-4 and Grid-5 essentially produces the same solution suggesting a grid-independence solution is obtained.

In Fig. 39 and Fig. #turbulent-airfoil-caelus-cfl3d-sacc-10 , the skin-friction is compared with CFL3D on Grid-4. As can be seen, a very good agreement between the two codes can be seen.

*Verification results: k-omega SST*

The skin-friction distribution obtained from using \(k-\omega~\rm{SST}\) turbulence model for \(\alpha=0^\circ\) and \(\alpha=10^\circ\) is shown below for different grids. The grid-sensitivity behaviour is very similar to the Spalart–Allmaras turbulence case and no change is seen between Grid-4 and Grid-5.

The comparison of the skin-friction with CFL3D using \(k-\omega~\rm{SST}\) is shown in Fig. 43 and Fig. 44 for both angle of attacks and similar to the previous case, a very good agreement between the two can be seen.

*Experimental validation*

Here, the Caelus data is compared with the pressure-coefficient (\(c_p\)) obtained experimentally by Gregory, N. and O’Reilly, C. L [5] for both angles of attack over the upper surface. In addition, the data obtained from CFL3D is also included for verification. There is a very good agreement with the current Caelus and experiments which indicates that the correct turbulence equations are being solved in both Spalart–Allmaras and \(k-\omega~\rm{SST}\) models.

**Conclusions**

Verification and validation over a two-dimensional NACA 0012 airfoil for turbulent inflow conditions were carried out using Caelus 4.10 employing simpleSolver. Two turbulence models that are implemented in-house were used and the solutions were verified with CFL3D data and subsequently validated with the experimental pressure coefficient values. The results were found to be in very good agreement suggesting that the turbulence modelling implementation is appropriate and solves accurately.

*Turbulent, incompressible flow in a two-dimensional convex curvature channel*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(a\) | Speed of sound | \(m/s\) |

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(c_p\) | Pressure coefficient | Non-dimensional |

\(I\) | Turbulent intensity | Percentage |

\(k\) | Turbulent kinetic energy | \(m^2/s^2\) |

\(M_i\) | Inlet Mach number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_L\) | Reynolds number | \(1/m\) |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(w\) | Local velocity in z-direction | \(m/s\) |

\(U\) | Inlet velocity | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(y\) | Distance in y-direction | \(m\) |

\(y^+\) | Wall distance | Non-dimensional |

\(\alpha\) | Angle of attack | Degrees |

\(\mu\) | Dynamic viscosity | \(kg/m~s\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\tilde{\nu}\) | Turbulence field variable | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

\(\omega\) | Specific dissipation rate | \(1/s\) |

\(i\) | Inlet conditions | |

\(t\) (subscript) | Turbulent property | |

\(ref\) | Reference pressure | \(Pa\) |

**Introduction**

In this case, the turbulent incompressible flow in a constant-area duct having a convex curvature is investigated as a part of verification and validation. The effect of curvature on the capability of turbulence model to predict the boundary layer accurately is of primary concern in this study. Similar to previous cases, the simulations are carried out over a series of four grids that tests the sensitivity of solution as the grid is refined. As with the earlier cases, Spalart-Allmaras (SA) and \(k-\omega~\rm{SST}\) turbulence models are tested. However due to the presence of strong curvature, a variant of Spalart–Allmaras turbulence model which accounts for rotational and curvature effects is additionally considered. This model is referred to as “Spalart–Allmaras Rotational/Curvature” (SA-RC) [9] . The results are then verified against CFL3D data and validated with experimental pressure distributions.

**Problem definition**

This verification and validation exercise is based on the Turbulence Modeling Resource case and follows the same conditions used in the incompressible limit. As CFL3D is a compressible CFD code, the original simulation used a inlet Mach number (\(M_i\)) of 0.093. A schematic of the geometric configuration is shown below and is depicted in \(x-z\) plane as the computational grid also follows the same 2D plane of flow.

As can be seen above, the 2D duct has a rapid bend of \(\alpha = 30^\circ\) after about a distance of 1.4 m and the downstream extends up to 1.6 m in length. The cross-section of the duct is 0.127 m and the inner radius and outer radius of curvature are 0.127 m and 0.254 m respectively. The flow has a Reynolds number of \(2.1 \times 10^6\) and U is the inlet velocity. To achieve a desired inlet velocity at an angle of 30 degrees, the velocity components are evaluated. The inlet temperature for this case is T = 293 K and for Air as a perfect gas, the speed of sound (\(a\)) can be evaluated to 343.106 m/s. Based on the inlet Mach number and speed of sound, the inlet velocity can be calculated to U = 31.908 m/s. Using velocity and Reynolds number kinematic viscosity can be calculated. The table below summarises the inlet conditions used:

Fluid | \(Re_L~(1/m)\) | \(U~(m/s)\) | \(p~(m^2/s^2)\) | \(T~~(K)\) | \(\nu_i~(m^2/s)\) |
---|---|---|---|---|---|

Air | \(2.1 \times 10^6\) | 31.9088 | \((0)\) Gauge | 293 | \(1.519470\times10^{-5}\) |

In order to achieve a inlet velocity of U = 31.9088 m/s at \(\alpha=30^\circ\), the velocity components in \(x\) and \(z\) are resolved. These are given below

\(\alpha~\rm{(Degrees)}\) | \(u~(m/s)\) | \(w~(m/s)\) |
---|---|---|

\(30^\circ\) | 27.63389 | 15.95443 |

It should be noted that in an incompressible flow, the density (\(\rho\)) does not vary and a constant density assumption is valid throughout the calculation. Further, the temperature field is not solved and therefore its influence on viscosity can be neglected and a constant viscosity can be used. In Caelus for incompressible flow simulations, pressure and viscosity are always specified as kinematic.

*Turbulent Properties for Spalart–Allmaras and Spalart–Allmaras Curvature Correction model*

The inflow conditions used for turbulent properties in Spalart-Allamaras model was calculated as \(\tilde{\nu}_{i} = 3 \cdot \nu_i\) and subsequently turbulent eddy viscosity was evaluated. The following table provides the values of these used in the current simulations:

\(\tilde{\nu}_i~(m^2/s)\) | \(\nu_{t~i}~(m^2/s)\) |
---|---|

\(4.558411 \times 10^{-5}\) | \(3.197543 \times 10^{-6}\) |

*Turbulent Properties for k-omega SST model*

\[k_{i} = \frac{3}{2} (U_i I)^2\]

\[\omega_{i} = 1 \times 10^{-6} \cdot \frac{\rho_i a^2_i}{\mu_i}\]

\[\nu_{t~i} = 0.009 \times \nu_i\]

Note that the dynamic viscosity in the above equation is obtained from Sutherland formulation and density is evaluated as \(\rho = \mu / \nu\). In the below table, the turbulent properties used in the current simulations are provided

\(I\) | \(k_{i}~(m^2/s^2)\) | \(\omega_{i}~(1/s)\) | \(\nu_{t~i}~(m^2/s)\) |
---|---|---|---|

\(0.083\%\) | \(1.0521 \times 10^{-3}\) | \(7747.333\) | \(1.36756 \times 10^{-7}\) |

**Computational Domain and Boundary Conditions**

The computational domain for the duct is quite simple and follows the geometry as is shown in Fig. 48. The walls are modelled as no-slip boundary and is highlighted in blue and the outlet is placed at the end of the duct.

*Boundary Conditions and Initialisation*

- Inlet
- Velocity: Fixed uniform velocity \(u = 27.63389~m/s\); \(v = 0.0~m/s\) and \(w = 15.95443~m/s\) in \(x, y\) and \(z\) directions respectively
- Pressure: Zero gradient
- Turbulence:
- SA & SA-RC (Fixed uniform values of \(\nu_{t~i}\) and \(\tilde{\nu}_{i}\) as given in the above table)
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k_{i}\), \(\omega_{i}\) and \(\nu_{t~i}\) as given in the above table)

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient
- Turbulence:
- SA & SA-RC (Fixed uniform values of \(\nu_{t}=0\) and \(\tilde{\nu} =0\))
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k = 0\) and \(\nu_t=0\); \(\omega\) = omegaWallFunction)

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)
- Turbulence:
- SA & SA-RC (Calculated \(\nu_{t}=0\) and Zero gradient \(\tilde{\nu}\))
- \(k-\omega~\rm{SST}\) (Zero gradient \(k\) and \(\omega\); Calculated \(\nu_t=0\); )

- Initialisation
- Velocity: Fixed uniform velocity \(u = 27.63389~m/s\); \(v = 0.0~m/s\) and \(w = 15.95443~m/s\) in \(x, y\) and \(z\) directions respectively
- Pressure: Zero Gauge pressure
- Turbulence:
- SA & SA-RC (Fixed uniform values of \(\nu_{t~i}\) and \(\tilde{\nu}_{i}\) as given in the above table)
- \(k-\omega~\rm{SST}\) (Fixed uniform values of \(k_{i}\), \(\omega_{i}\) and \(\nu_{t~i}\) as given in the above table)

**Computational Grid**

The computational grid in 3D for the convex curvature duct was obtained from Turbulence Modeling Resource as a Plot3D format. The same was used in Caelus by converting it in the required format with the use of Pointwise. Since the flow field is assumed to the two-dimensional, the 2D computational plane of interest is in \(x-z\) directions. SimpleSolver is a 3D solver, therefore the two additional \(x-z\) planes are specified with empty boundary conditions. To look at the effect of grid sensitivity, four grids were considered from the original set of five, while the coarsest grid was not included in this study. The table below gives the details of different grids used.

Grid | Cells in streamwise direction | Cells in normal direction | Total |
---|---|---|---|

Grid-2 | 128 | 48 | 6144 |

Grid-3 | 256 | 96 | 98304 |

Grid-4 | 512 | 192 | 98,304 |

Grid-5 | 1024 | 384 | 393,216 |

The 2D convex curvature grid is shown in Fig. 49 below, for Grid-3 in \(x-z\) plane and the inset shows the grids in the vicinity of the strong curvature. Grids are refined close to the wall in order to capture the turbulent boundary layer and all grids have a \(y^+ < 1\) and no wall function is used through the validation and verification of this configuration.

**Results and Discussion**

The turbulent flow inside the convex curvature duct was simulated using Caelus 4.10 through the use of simpleSolver. The solutions were run until the residuals for pressure, velocity and turbulent quantities were less than \(1 \times 10^{-6}\). The finite volume discretization of the gradient of pressure and velocity was carried out using the linear approach. Where as the divergence of velocity and mass flux was carried out through the linear upwind method. However, for the divergence of the turbulent quantities, upwind approach was utilised and linear approach for the divergence of the Reynolds stress terms. For the discretization of the Laplacian terms, again linear corrected method was used. For some grids having greater than 50 degree non-orthogonal angle, linear limited with a value of 0.5 was used for the Laplacian of the turbulent stress terms.

The verification results of the turbulence model are discussed first, which is then followed by the experimental validation.

*Verification results Spalart–Allmaras (SA)*

In Fig. 50, the skin-friction distribution obtained over the lower wall of the duct is shown from Caelus for different grids. As can be seen, there is very little difference in skin-friction variation among the different grids. The oscillatory behaviour noted at Grid-5 very close to the corner is also apparent in CFL3D data as will be see in Fig. 50.

The skin-friction coefficient comparison with CFL3D is shown in Fig. 51. It should be noted that the available solution from CFL3D was for Grid-4 and hence to be consistent, Grid-4 solution from Caelus is used for comparison. In the vicinity of strong curvature region, Caelus compares very well with CFL3D, however both upstream and downstream, there seems to be some difference in the solution.

*Verification results Spalart–Allmaras Rotational/Curvature (SA-RC)*

The following Fig. 52 shows the grid sensitivity and verification with CFL3D data respectively. The solution that have been used for verification is obtained from Grid-4 and the trends are similar to what is noted for the SA model.

*Verification results k-Omega SST*

In Fig. 54, the skin-friction sensitivity is shown over the lower wall obtained using Caelus with the \(k-\omega~\rm{SST}\) model. After Grid-3, not much difference in values can be noted. With Grid-5, however some oscillations can be see upstream and in the vicinity of the curvature.

The skin-friction comparison between Caelus and CFL3D is shown in Fig. 55. A very good agreement between the two is obtained.

*Experimental validation*

This section details the experimental validation carried out for Caelus and both skin-friction and pressure coefficients obtained experimentally by Smits, A. J et al. [8] are compared. Further, CFL3D is also included. In Fig. 56, skin-friction distribution obtained from Caelus using different turbulence models is compared with the experiments. Both SA-RC and \(k-\omega~\rm{SST}\) has a fair agreement with experiments down stream of the curvature, more so with the SA-RC model. However upstream they all seem to predict nearly the same values.

Fig. 57 shows the comparison of pressure-coefficient (\(c_p\)) distribution with experiments and CFL3D data on the lower surface. Firstly, the solutions obtained from Caelus with the three turbulence models essentially produces the same values and matches exactly with the CFL3D data. In comparison with experiments, the agreement is very good in the upstream, vicinity of the curvature and downstream and, identical to CFL3D’s behaviour. Note that for obtaining the pressure-coefficient (\(c_p\)) values, a reference pressure (\(p_{ref}\)) is needed. However, this is not specified in Turbulence Modeling Resource for this case and hence a value of 145 Pa has been used.

Experimental skin-friction data [8] is also available over the upper surface of the duct and is used to compare the Caelus results. Fig. 58 shows the comparison. Similar to the behaviour noted for the lower surface, the SA-RC model tends to be closer to the experimental data. In general, all the three turbulence model have similar trends and agrees very closely with the CFL3D data.

**Conclusions**

A detailed verification and validation of a turbulent flow in a convex curvature duct were carried out using Caelus 4.10 and simpleSolver. Here, three turbulence models used and the solutions were verified against CFL3D data. As a part of validation, Caelus results were compared with the experimental data obtained on both lower and upper surfaces. The comparison was good with both CFL3D as well as with experiments. This suggests that the implementation of the turbulence models is correct and is being solved accurately.

*Turbulent, incompressible flow over a two-dimensional backward facing step*

**Nomenclature**

Symbol | Definition | Units (SI) |
---|---|---|

\(a\) | Speed of sound | \(m/s\) |

\(c_f\) | Skin friction coefficient | Non-dimensional |

\(c_p\) | Pressure coefficient | Non-dimensional |

\(H\) | Step height | \(m\) |

\(I\) | Turbulent intensity | Percentage |

\(k\) | Turbulent kinetic energy | \(m^2/s^2\) |

\(M_\infty\) | Freestream Mach number | Non-dimensional |

\(p\) | Kinematic pressure | \(Pa/\rho~(m^2/s^2)\) |

\(Re_H\) | Reynolds number based on step height | Non-dimensional |

\(T\) | Temperature | \(K\) |

\(u\) | Local velocity in x-direction | \(m/s\) |

\(u_*\) | Frictional velocity | \(m^2/s\) |

\(U\) | Freestream velocity in x-direction | \(m/s\) |

\(U_{ref}\) | Reference velocity | \(m/s\) |

\(x\) | Distance in x-direction | \(m\) |

\(z\) | Distance in z-direction | \(m\) |

\(y^+\) | Wall distance | Non-dimensional |

\(\epsilon\) | Turbulent dissipation | \(m^2/s^3\) |

\(\mu\) | Dynamic viscosity | \(kg/m~s\) |

\(\nu\) | Kinematic viscosity | \(m^2/s\) |

\(\tilde{\nu}\) | Turbulence field variable | \(m^2/s\) |

\(\rho\) | Density | \(kg/m^3\) |

\(\tau_w\) | Wall shear stress | \(kg/m~s^2\) |

\(\omega\) | Specific dissipation rate | \(1/s\) |

\(\infty\) | Freestream conditions | |

\(t\) (subscript) | Turbulent property |

**Introduction**

This study investigates steady turbulent, incompressible flow over a two-dimensional backward facing step at zero angle of incidence. Unlike the previous cases, the efficacy of wall functions in separated flow is evaluated for different turbulent models. The validation of these wall functions are carried out through the comparison of skin-friction coefficient (\(c_f\)) and pressure coefficient (\(c_p\)) downstream of the step with those of the experimental data. In addition to Spalart–Allmaras and \(k - \omega~\rm{SST}\), Realizable \(k-\epsilon\) turbulence model was also considered in this exercise.

**Problem definition**

The backward facing step configuration is obtained from the Turbulence Modeling Resource and is a widely considered case for the purpose of verification and validation. This particular study is based on the experiments carried out by Driver and Seegmiller [15]. The schematic of the step configuration in Fig. 59 below as considered in the Turbulence Modeling Resource.

The height of the step (H) is chosen to be 1.0 m and is located at x = 0 m. Upstream of the step, the plate extends to 110 step heights such that a fully developed turbulent boundary layer with proper thickness exists at separation (x = 0 m). This is followed by a downstream plate which extends up to 50 step heights. The flow has a Reynolds number of 36000 based on the step height with a freestream velocity (\(U_\infty\)) in the x-direction. The inflow is assumed to be Air as a perfect gas with a temperature of 298 K, giving the value of speed of sound (\(a\)) at 346.212 m/s. A reference Mach number (\(M_\infty\)) of 0.128 is used to obtain the freestream velocity, which was \(U_\infty\) = 44.315 m/s. The kinematic viscosity was then evaluated based on the velocity and the Reynolds number. The following table summarises the freestream conditions used for the backward facing step.

Fluid | \(Re_H\) | \(U_\infty~(m/s)\) | \(p_\infty~(m^2/s^2)\) | \(T_\infty~(K)\) | \(\nu_\infty~(m^2/s)\) |
---|---|---|---|---|---|

Air | \(36000\) | 44.31525 | \((0)\) Gauge | 298.330 | \(1.230979\times10^{-3}\) |

As with all the previous cases, the flow is incompressible and therefore the density (\(\rho\)) does not change throughout the simulation. Further, the temperature is not accounted and has no influence on viscosity and is also held constant. However note that in Caelus, pressure and viscosity for incompressible flow are always specified as kinematic.

*Turbulent Properties for Spalart–Allmaras model*

The turbulent boundary conditions at the freestream used for Spalart-Allmaras model were calculated according to \(\tilde{\nu}_{\infty} = 3 \cdot \nu_\infty\) and turbulent eddy viscosity was subsequently evaluated. In the following table, the values used in the current simulation are provided:

\(\tilde{\nu}_\infty~(m^2/s)\) | \(\nu_{t~\infty}~(m^2/s)\) |
---|---|

\(3.692937 \times 10^{-3}\) | \(2.590450 \times 10^{-4}\) |

*Turbulent Properties for k-omega SST model*

\[k_{\infty} = \frac{3}{2} (U_\infty I)^2\]

\[\omega_{\infty} = 1 \times 10^{-6} \cdot \frac{\rho_\infty a^2_\infty}{\mu_\infty}\]

\[\nu_{t~\infty} = 0.009 \times \nu_\infty\]

The dynamic viscosity in the above equation is obtained from Sutherland formulation and the density is calculated through \(\mu/\nu\). In the table below, the turbulent properties used for the current \(k-\omega~\rm{SST}\) simulation are provided

\(I\) | \(k_{\infty}~(m^2/s^2)\) | \(\omega_{\infty}~(1/s)\) | \(\nu_{t~~\infty}~(m^2/s)\) |
---|---|---|---|

\(0.061\%\) | \(1.0961 \times 10^{-3}\) | \(97.37245\) | \(1.10787 \times 10^{-5}\) |

*Turbulent Properties for Realizable k-epsilon model*

The turbulent inflow properties used for Realizable \(k-\epsilon\) model were evaluated as follows

\[k_{\infty} = \frac{3}{2} (U_\infty I)^2\]

\[\epsilon_{\infty} = \frac{0.1643~k_{\infty}^{3/2}}{\lambda}\]

\[\nu_{t\infty} = \frac{C_\mu~k^2}{\epsilon_{\infty}}\]

where, \(\lambda\) is the turbulent length scale and is evaluated at 0.22 of the boundary layer thickness at separation (1.5H) and \(C_\mu\) is a model constant with a value 0.09. The following table provides the evaluated turbulent properties. Note that the turbulent intensity was assumed to be 1 % for this particular model.

\(I\) | \(k_{\infty}~(m^2/s^2)\) | \(\epsilon_{\infty}~(m^2/s^3)\) | \(\nu_{t\infty}~(m^2/s)\) |
---|---|---|---|

\(1\%\) | \(294.57 \times 10^{-3}\) | \(0.079598\) | \(98.11 \times 10^{-3}\) |

**Computational Domain and Boundary Conditions**

The computational domain simply follows the step geometry for the entire bottom region. In Fig. 60 below, the boundary details in two-dimensions (\(x-z\) plane) are shown. The walls of the upstream plate, step and the downstream plate that extend between \(-110 \leq x \leq 50~m\) are modelled as no-slip wall boundary condition. Similarly, the top plate is also modelled as a no-slip wall. Upstream of the leading edge, that is, \(x \leq 110\) symmetry boundary extends for a length of 20 step heights and the inlet boundary is placed at the start of the symmetry. The outlet is placed at the end of the downstream plate, which is at \(x = 50~m\).

*Boundary Conditions and Initialisation*

- Inlet
- Velocity: Fixed uniform velocity \(u = 44.31525~m/s\) in \(x\) direction
- Pressure: Zero gradient
- Turbulence:
- Realizable \(k-\epsilon\) (Fixed uniform value of \(k_{\infty}\), \(\epsilon_{\infty}\) and \(\nu_{t_\infty}\) as given in the above table)

- Symmetry
- Velocity: Symmetry
- Pressure: Symmetry
- Turbulence: Symmetry

- No-slip wall
- Velocity: Fixed uniform velocity \(u, v, w = 0\)
- Pressure: Zero gradient
- Turbulence:
- Spalart–Allmaras:
- \(\nu_t\): type nutUWallFunction with an initial value of \(\nu_t=0\)
- \(\tilde{\nu}\): type fixedValue with a value of \(\tilde{\nu}=0\)

- \(k-\omega~\rm{SST}\):
- \(k\): type kqRWallFunction with an initial value of \(k_{\infty}\)
- \(\omega\): type omegaWallFunction with an initial value of \(\omega_{\infty}\)
- \(\nu_t\): type nutUWallFunction with an initial value of \(\nu_t=0\)

- Realizable \(k-\epsilon\):
- \(k\): type kqRWallFunction with an initial value of \(k_{\infty}\)
- \(\epsilon\): type epsilonWallFunction with an initial value of \(\epsilon=0\)
- \(\nu_t\): type nutUWallFunction with an initial value of \(\nu_t=0\)

- Spalart–Allmaras:

- Outlet
- Velocity: Zero gradient velocity
- Pressure: Fixed uniform gauge pressure \(p = 0\)
- Turbulence:
- Spalart–Allmaras (Calculated \(\nu_{t}=0\) and Zero gradient \(\tilde{\nu}\))
- \(k-\omega~\rm{SST}\) (Zero gradient \(k\) and \(\omega\); Calculated \(\nu_t=0\); )
- Realizable \(k-\epsilon\) (Zero gradient \(k\) and \(\epsilon\); Calculated \(\nu_t=0\); )

- Initialisation
- Velocity: Fixed uniform velocity \(u = 44.31525~m/s\) in \(x\) direction
- Pressure: Zero Gauge pressure
- Turbulence:
- Realizable \(k-\epsilon\) (Fixed uniform values of \(k_{\infty}\), \(\epsilon_{\infty}\) and \(\nu_{t~\infty}\) as given in the above table)

**Computational Grid**

The 3D computational grid was generated using Pointwise. and was converted to Caelus format. Note that the plane of interest is in \(x-z\) directions. Further, since the flow field is assumed to be two-dimensional, the two \(x-z\) obtained as a result of a 3D grid are specified with empty boundary condition. This forces the 3D solver, simpleSolver to treat the flow in \(y\) direction as symmetry. The following figure shows the 2D grid. As noted earlier, the purpose of this exercise is to validate the wall functions and hence the grid is designed for a \(y^+ = 30\). In order to design such a gird, the first cell height (\(\Delta z\)) from wall in the wall normal (\(z\)) direction was obtained from the following set of equations

\[\Delta z = \frac{y^+~\nu}{u_*}\]

where, \(u_*\) is the frictional velocity given by

\[u_* = \sqrt{\frac{\tau_w}{\rho}}\]

In the above equation, \(\tau_w\) is the shear-stress at the wall and was be estimated using the skin-friction (\(c_f\)) relation, given as

\[\tau_w = c_f\frac{1}{2} \rho u_\infty^2\]

where, \(c_f\) was obtained for the flat-plate as given in Schlichting [7] and is shown below

\[c_f = [2~\textrm{log}(Re_x) - 0.65]^{-2.3}\]

where, \(Re_x\) is the Reynolds number based on the length of the boundary layer. In this case, it is the length developed over the upstream plate.

The 2D grid of a backward facing step in \(x-z\) is shown in Fig. 61 for a \(y^+ \approx 30\). In the upstream region of the step, there are 60 cells in the streamwise and 64 in the wall normal directions respectively. Downstream of the step, there are 129 cells in the streamwise and a total of 84 cells in the normal direction. Out of 84 cells, 20 cells represent the height of the step.

**Results and Discussion**

The steady-state solution of turbulent flow over a two-dimensional backward facing step was obtained using Caelus 4.10. The simpleSolver was used and the simulation was run sufficiently long until the residuals for pressure, velocity and turbulent quantities were less than \(1 \times 10^{-6}\). The finite volume discretization of the gradient of pressure and velocity was carried out using linear approach. Where as the divergence of velocity and mass flux was carried out through the linear upwind method. However, for the divergence of the turbulent quantities, upwind approach was utilised. For the discretization of the Laplacian terms, again linear method was used.

*Experimental validation of skin-friction coefficient*

In this section, the validation carried out for Caelus based on skin-friction and pressure obtained experimentally by Driver and Seegmiller [15] is presented. The results obtained from CFL3D [1] are additionally shown and should be considered only as a reference and not as a benchmark for verification. This is because all the CFL3D results have been obtained without the use of wall functions and on a grid having \(y^+ \approx 1\). In Fig. 62, skin-friction distribution obtained from Caelus using SA turbulence model is compared with the experiments. Upstream of the step, the agreement is good, however, downstream post-reattachment the skin-friction under predicts the experimental data. In both these regions of the flow, Caelus results are nearly identical to that of CFL3D suggesting that the wall-function is capturing the flow characteristics accurately. Within the separated region, there is a large discrepancy and this is due to the inherent low \(y^+\) mesh in that region, where typically a wall function becomes invalid.

Figure Fig. 63 gives the comparison of skin-friction obtained from \(k-\omega~\rm{SST}\) turbulence model. The result is very similar to the one obtained from SA model. In this case, the skin-friction upstream of the step is slightly under predicted, whereas, post reattachment, it seems to be closer to experiments. In contrast with the SA result, the skin-friction is now closer to the experimental data within the separated region, particularly in the region closer to the reattachment location.

In Fig. 64, the comparison is shown for Realizable \(k-\epsilon\) turbulence model. Note that CFL3D data was not available for this turbulence model to use as a reference. Again, similar skin-friction behaviour can be noted both upstream and downstream of the step with reasonable agreement with the experimental data. Within the separated region, there is a large difference and this could be due to the presence of low \(y^+\) mesh as discussed earlier.

One of the key feature of modelling the backward facing step is the accurate prediction of reattachment location downstream of the step. This was determined through the location at which the reversal of skin-friction occurs over the downstream surface for each of the turbulence model considered here. In the below table, the normalised reattachment distances obtained from Caelus simulation are compared with the experimental data. Out of the three models considered here, Realizable \(k-\epsilon\) prediction is in very good agreement with the experimentally obtained value.

Type | Reattachment location (\(x/H\)) |
---|---|

Experimental | \(6.26 \pm 0.10\) |

SA | \(5.55\) |

\(k-\omega~\rm{SST}\) | \(6.08\) |

Realizable \(k-\epsilon\) | \(6.27\) |

*Experimental validation of pressure coefficient*

Fig. 65 gives the pressure-coefficient (\(c_p\)) comparison among three Caelus simulations and the experimental data. The inclusion of CFL3D data is again only for reference and not as a benchmark comparison. All the simulations essentially produce the same trend and is consistent with the skin-friction coefficient distribution. The pressure prediction in both \(k-\omega~\rm{SST}\) and Realizable \(k-\epsilon\) are very close to each other over the entire region shown in the figure and is also in fair agreement with the experimental data. However, SA seems to show some significant deviation particularly in the region of pressure minima.

**Conclusions**

A detailed validation of the turbulent flow over a backward facing step was carried out using Caelus 4.10 for a simpleSolver. In particular the focus was to validate wall functions for grids having \(y^+ \approx 30\). The solutions from three turbulence models were compared with both skin-friction and pressure coefficients obtained experimentally over the surface of the model. Overall, a good agreement was noted. With respect to the reattachment location, the simulated result with Realizable \(k-\epsilon\) turbulence model agreed very close to the experimental data. Considering both the skin-friction and pressure data, the implementation of the turbulence model is correct with providing accurate solutions.