Numerical analysis of flow over Ahmed body using ANSYS fluent

Objective:

1. To perform numerical analysis on flow over Ahmed body and to perform grid dependence test.

Ahmed body and its importance:

Ahmed body has the form of a highly simplified car, consisting of a blunt nose with rounded edges fixed onto a box-like middle section and a rear end that has an upper slanted surface (like a “hatch-back” car), the angle of which can be varied. The model is supported on circular-sectioned legs or stilts, rather than wheels. Despite neglecting a number of features of a real car (rotating wheels, rough underside, surface projections etc.) the Ahmed body generates the essential features of flow around a car, namely: flow impingement and displacement around the nose, relatively uniform flow around the middle and flow separation and wake generation at the rear.

The Ahmed body was first created by S.R.Ahmed in his research \"some salient of the time averaged ground vechicle wake\" in 1984.Since then it has become a benchmark for aerdynamic simulation tools.The principal aim of studying such a simplified car body is to understand the flow processes involved in drag production. Through understanding the mechanisms involved in generating drag one should be able to design a car to minimize drag and therefore minimize fuel consumption and maximize performance.Ahmed body helps to demontrate how the drag of a body is mainly due to the pressure drag generated at the rear of the body.

Computational flow domain:

In order to perform analysis of external flow over ahmed body, outer enclosure is created and inorder to refine the mesh near the surface of the ahmed body(so that the flow can be captured accurately) inner enclosure is created.

Dimensions of outer enclosure:

Dimensions are measured from origin located at the front of ahmed body as shown in figure above.

Distance to inlet(from origin)(cushion -x)= 2 m

Distance to outlet(cushion +x)= 5m

Side walls(cushion -z, +z)= 0.5 m

Top wall( cushion +Y)= 1 m

base(cushion -Y)= 0 m.

Dimensions of inner enclosure:

Dimensions are measured from origin located at the front of ahmed body as shown in figure above.

Cushion -x= 0.5 m

Cushion +x= 1 m

Cushion -Y= 0 m

Cushion +Y= 0.5 m

Cushion +z= 0.26 m

Cushion -z= 0.26 m

Fluid properties:

Fluid: Air

Density(kg/m^3) : 1.225

Dynamic viscosity(kg/m-s) : 1.7894e-05

Cp(j/kg-K) : 1006.43

Reynolds number and Mach number calculation:

Density of fluid= 1.225

Velocity of fluid at inlet(V)= 50 m/s

Characteristic length at inlet(D) = 1 m

Dynamic viscosity(kg/m-s) : 1.7894e-05

Velocity of sound in air= 334 m/s

Reynolds number,

`Re= frac{rhoVD}{mu}=frac{1.225*50*1}{0.000017894}=3422935`, Hence turbulent flow, Therefore `k-epsilon` turbulence model is adopted for our problem.

Mach number,

Mach number is the ratio of velocity of fluid flow to the local speed of sound in fluid,

`M=frac{50}{343}= 0.1457`, Hence incompressible flow(M<0.3), In order to capture compressiblity if there is any, Density based solver is used for our problem.

Governing Equations:

Continuity equation:

`frac{delrho}{delt}+ nabla.(rhoV)=0`

Momentum equation:

x component: `frac{del(rhou)}{delt}+grad.(rhouV)=-frac{delp}{delx}+frac{del(tau_(x x ))}{delx}+frac{del(tau_(yx))}{dely}+frac{del(tau_(zx))}{delz}+rhof_x`

Y component: `frac{del(rhov)}{delt}+grad.(rhovV)=-frac{delp}{dely}+frac{del(tau_(x y ))}{delx}+frac{del(tau_(yy))}{dely}+frac{del(tau_(zy))}{delz}+rhof_y`

z component: `frac{del(rhow)}{delt}+grad.(rhowV)=-frac{delp}{delz}+frac{del(tau_(x z ))}{delx}+frac{del(tau_(yz))}{dely}+frac{del(tau_(zz))}{delz}+rhof_z`

For current case, we have considered steady, in-compressible flow.

Turbulence Modeling:

The Reynolds-averaged Navier-Stokes (RANS) equations govern the transport of the averaged flow quantities, with the whole range of the scales of turbulence being modeled. The RANS-based modeling approach therefore greatly reduces the required computational effort and resources, and is widely adopted for practical engineering applications.

Standard `k-epsilon` Turbulence Model:

The standard k-ε model is a model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (ε). The model transport equation for k is derived from the exact equation, while the model transport equation for ε was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.

Transport Equations for Standard `k-epsilon` Turbulence Model:

The modelled transport equations for k and `epsilon` in the Standard `k-epsilon` model are

`frac{del(rhok)}{delt}+frac{del(rhoku_j)}{delx_j}=frac{del}{delx_j}[(mu+frac{mu_t}{sigma_k})frac{delk}{delx_j}]+G_k+G_b-rhoepsilon-Y_M+S_k`

Physical Meaning of k equation:

Time derivative of k+ Convective transport = diffusive transport + Rate of production-Rate of destruction

`frac{del(rhoepsilon)}{delt}+frac{del(rhoepsilonu_j)}{delx_j}=frac{del}{delx_j}[(mu+frac{mu_t}{sigma_(epsilon)})frac{del(epsilon)}{delx_j}]+C_(1epsilon)frac{epsilon}{k}(G_k+C_(3epsilon)G_b)-C_(2epsilon)rhofrac{epsilon^2}{k}+S_epsilon`

Physical Meaning of `epsilon` equation:

Time derivative of ε+ Convective transport = Diffusive transport + Rate of production-Rate of destruction.

where,

k= Turbulent kinetic energy

`epsilon`= Turbulent dissipation rate

`G_k`= generation of turbulence kinetic energy due to the mean velocity gradients

`G_b`= generation of turbulence kinetic energy due to buoyancy

`Y_M`= represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate

`sigma_k`,`sigma_epsilon`= turbulent Prandtl numbers for k and `epsilon` respectively

`C_(1epsilon),C_(2epsilon),C_(3epsilon)`= Constants

Model Constants:

Boundary Conditions:

Suitable boundary conditions have been applied according to the physics of the flow.

1. At Inlet :

Velocity inlet of magnitude 50 m/s normal to boundary,

Turbulent intensity(%)= 5 and turbulent viscosity ratio= 10.

2. At outlet:

Pressure outlet at zero gauge pressure(atmospheric pressure),

Turbulent intensity(%)= 5 and turbulent viscosity ratio= 10.

3. At symmetry: (Side walls and top wall of outer enclosure)

Symmetry boundary condition is applied.

Symmetry Boundar condition:The symmetry boundary condition defines a mirror face/surface. The symmetry condition should only be used if the physical object or geometry and the expected flow field pattern of the developed solution is mirrored along that surface. This boundary condition helps to reduces the computational domain in size and enables the modeling of a sub-domain of the complete setup.

In general, the symmetry condition applies the following constraints on the flow variables:

• The fluxes across a symmetry are zero.

• The normal components of velocity are set to zero.

• The normal components of all other variables are set to zero.

4. At base: (bottom of the enclosure)

Stationary wall with no slip condition(velocity of fluid which is in contact with wall surface is zero)

5. At body_wall: (Ahmed body)

Stationary wall with no slip condition(velocity of fluid which is in contact with wall surface is zero)

Solver schemes:

1. Solver: Density based

2. Formulation: Implicit

3. Flux type: Roe-FDS

Density based solver: The density-based solver solves the governing equations of continuity, momentum, and (where appropriate) energy and species transport simultaneously (i.e., coupled together).Because the governing equations are non-linear (and coupled), several iterations of the solution loop must be performed before a converged solution is obtained.In the density-based solution methods, the discrete, non-linear governing equations are linearized to produce a system of equations for the dependent variables in every computational cell. The resultant linear system is then solved to yield an updated flow-field solution.

The manner in which the governing equations are linearized may take an \"implicit\'\' or \"explicit\'\' form with respect to the dependent variable (or set of variables) of interest.

Implicit: For a given variable, the unknown value in each cell is computed using a relation that includes both existing and unknown values from neighboring cells. Therefore each unknown will appear in more than one equation in the system, and these equations must be solved simultaneously to give the unknown quantities.

Roe-FDS: Roe-FDS splits the fluxes in a manor that is consistent with their corresponding flux method eigenvalues. It is the default and is recommended for most cases.

4. Spatial discretization:

Gradient: Least squares cell based

Flow: Second order upwind

Turbulent kinectic energy: First order upwind

Turbulent dissipation energy : First order upwind

First-order upwind scheme- When first-order accuracy is desired, quantities at cell faces are determined by assuming that the cell-center values of any field variable represent a cell-average value and hold throughout the entire cell; the face quantities are identical to the cell quantities.

Second-order upwind scheme- When second-order accuracy is desired, quantities at cell faces are computed using a multidimensional linear reconstruction approach . In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid.

Initialization:

Hybrid Initialization: Hybrid Initialization is a collection of recipes and boundary interpolation methods. It solves the Laplace equation to produce a velocity field that conforms to complex domain geometries, and a pressure field which smoothly connects high and low pressure values in the computational domain. All other variables (that is temperature, turbulence, VOF, species, and so on) will be patched based on domain averaged values or a predetermined recipe.

Mesh:

Case 1:

Case 2:

Case 3:

The sectional view is shown above for case 1,2 and 3.All all three cases, Inner-enclosure is created in order to capture the flow physics accurately and also to save computational time.In order to refine the mesh size inside the inner enclosure body sizing is used.Since there is no curved surface between the outer enclosure and inner enclosure, Mutli-zone is used to mesh the required domain with hexahedral mesh.The circular faces of the ahmed body supports is refined by using face sizing. In order to capture the boundary layer between the ahmed body and fluid, Inflation is used in named selection body_wall.

Residuals:

Case 1:

Case 2:

Case 3:

From above plots, we can observe that the residuals for case 1,2 and 3 have decreased and solution have been converged.The convergence criteria used for all the three cases is 1e-03.

Results:

Case 1:

1. Pressure contour at cut_plane:

2. Pressure contour at body_wall(ahmed body):

3. Velocity contour at cut_plane:

4. Velocity vector at cut_plane:

5. Drag coefficient plot for every iteration:

6. Velocity Distribution at cross section 3.2 m from inlet:

Case 2:

1. Pressure contour at cut_plane:

2. Pressure contour at body_wall(ahmed body):

3. Velocity contour at cut_plane:

4. Velocity vector at cut_plane:

5. Drag coefficient plot for every iteration:

6. Velocity Distribution at cross section 3.2 m from inlet:

Case 3:

1. Pressure contour at cut_plane:

2. Pressure contour at body_wall(ahmed body):

3. Velocity contour at cut_plane:

4. Velocity vector at cut_plane:

5. Drag coefficient plot for every iteration:

6. Velocity Distribution at cross section 3.2 m from inlet:

Discussions:

1. By observing the pressure contours at cut_plane and body_wall, There is an increase in pressure at the frontal area of the ahmed body and therefore there is a decrease in pressure(negative pressure) near the top edge and near the rear slant of the ahmed body. This may be due to increase in velocity in those specified regions and also it can be observed that wherever there is decrease in pressure near the regions of flow over ahmed body, there is an increase in velocity at those regions.

2.Adverse pressure gradient can be observed in the pressure contour on ahmed body.Also flow separation of the boundary layer can be observed in the wake region(backside) of the ahmed body.

3.Adverse pressure gradient: Adverse pressure gradient is one in which pressure increases in the flow direction (i.e., dp/dx > 0); it is called adverse because it will cause fluid particles in the boundary-layer to slow down at a greater rate than that due to boundary-layer friction alone.For a large enough pressure increase, this fluid may slow to zero velocity or even become reversed causing a flow separation. This has very significant consequences in aerodynamics since flow separation significantly modifies the pressure distribution along the surface and hence the lift and drag characteristics.

4. Pressure Drag: The boundary layer of air moving over the body(ahmed body) slows down because it is forced to move from an area of low pressure to an area of high pressure(adverse pressure gradient). When the boundary layer of air slows down to a certain point, it detaches from the surface of the body and becomes turbulent. This turbulent air has a lower pressure, which has an effect on pressure drag. Pressure drag occurs when air flowing past an object pushes harder against the front than the back. This difference creates a backward force. When separation of flow occurs, the turbulent air behind the airfoil decreases in pressure, causing an increase in pressure drag.

5. Flow separation: Flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake.The boundary layer separates when it has travelled far enough in an adverse pressure gradient that the speed of the boundary layer relative to the surface has stopped and reversed direction.The flow becomes detached from the surface, and instead takes the forms of eddies and vortices.In aerodynamics, flow separation results in reduced lift and increased pressure drag, caused by the pressure differential between the front and rear surfaces of the object.

6. The flow separation occuring in the wake region and formation vortices in the wake region can be seen in detailed in the velocity vector contours of case 2 and case 3.

7. The drag coefficient obtained in case 1 is 111.9 , drag coefficient for case 2 is 98.2215 and for case 3 is 102.108. Therefore considering the accuray of the solution( detailing of velocity vector in wake region, drag coefficient) and computational time, Solution of case 2(number of elements=353699) can be considered as grid independent.

8. The principal contribution to drag experienced by a car is pressure drag. The rear of the vehicle provides the major contribution to pressure drag and, in particular, the angle of the rear slant is critical in determining the mode of the wake flow and hence the drag experienced by the vehicle.