COMPUTATIONAL FLUID DYNAMICS

COMPUTATIONAL FLUID DYNAMICSABSTRACT The sleek & beautiful aircraft roles down the run away, takes off & rapidly climbs out of sight within a minute, this same aircraft has accelerated to hypersonic speed; still within atmosphere. Its powerful supersonic engine continues to propel aircraft with velocity near 26000 ft/s orbital velocity and vehicle simply coasts into low earth orbit. Is this the stuff of dreams? Not really; indeed this is possible due to CFD. CFD is the art of replacing the differential equation governing the fluid flow, with a set of algebraic equations (the process is called discretization). Which in turn can be solved with aid of digital computer to get an approximate solution.The physical aspects of any fluid flow are governed by three fundamental principles (1) Mass is conserved (continuity equation) (2)Newtons second law (Navier-Stokes equations)(3) Energy is conserved (energy equation) CFD is a research tool & playing a strong role as design tool, in aerodynamics as well as non-aerospace applications like automobile, engines (flow field inside I.C. engines ), industrial manufacturing (actual behavior of molten metal in mould ), civil & environmental engg. In a next decade CFD will become a critical technology for aerodynamic design & on other hand concurrent engg. is possible by drastic changes, shortening of design process & optimization of air vehicle system in terms of overall economical performance. CFD is growth industry with an unlimited numbernos of new application & new ideas just waiting in future. This paper includes Dicretization , Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Element Method (FEM), Boundary Element Method (BEM), Grid(mesh generation), Navier Stokes Equations & greater advantage of CFD over wind tunnel testing. This paper also highlights on AERODYNAMIC DESIGN OF FORMULA ONE RACING CAR BY CFD (simulation technique)Index1. Introduction[1]2. Discretization[2]3. Navier stokes equations[10] 4. Wind Tunnel Testing V/s CFD [12]5. Aerodynamic Analysis of Formula one racing car by CFD[13]6. Applications[18]7. Limitation[18]8. Conclusion[19]Reference

COMPUTATIONAL FLUID DYNAMICS1. Introduction Computational Fluid Dynamics (CFD) has grown from a mathematical curiosity to become an essential tool in almost every branch of fluid dynamics, from aerospace propulsion to weather prediction. CFD is commonly accepted as referring to the broad topic encompassing the numerical solution, by computational methods, of the governing equations which describe fluid flow, the set of the Navier-Stokes equations, continuity and any additional conservation equations, for example energy or species concentrations. What do we mean when we speak of simulating a fluid flow on a computer? In simplest terms, the computer solves a series of well-known equations that are used to compute, for any point in space near an object, the velocity and pressure of the fluid flowing around that object. Computational Fluid Dynamics or simply CFD is concerned with obtaining numerical solution to fluid flow problems by using computers. The advent of high-speed and large-memory computers has enabled CFD to obtain solutions to many flow problems including those that are compressible or incompressible, laminar or turbulent, chemically reacting or non-reacting.

2.DISCRETIZATION CFD is the art of replacing the differential equation governing the Fluid Flow, with a set of algebraic equations (the process is called discretization), which in turn can be solved with the aid of a digital computer to get an approximate solution The well-known discretization methods used in CFD are: - Finite Difference Method (FDM) Finite Volume Method (FVM) Finite Element Method (FEM) Boundary Element Method (BEM) FINITE DIFFERENCE METHOD: - . Here the domain including the boundary of the physical problem is covered by a grid or mesh. At each of the interior grid point the original Differential Equations are replaced by equivalent finite difference approximations. In making this replacement, we introduce an error, which is proportional to the size of the grid. Making the grid size smaller to get an accurate solution within some specified tolerance can reduce this error. FINITE VOLUME METHODFinite volume method

Consider a domain in which fluid is flowing. Solution to flow problem constitutes knowing flow variables in the domain at as many points as possible. More the number of points more accurate will be the solution but higher will be the cost of computation.Let us assume that domain of the flow is divded in many small triangles(or similar such shapes),such that collectively they cover the whole domain;but they do not overlap each other.Each triangle is called an element.More on element is given in section on Grid Generation.

The first terms in above equations give the rate at which(1) mass,(2)momentum in x-direction, and (3)momentum in y-direction in a given triangle increases with time.The second term in the first equation gives the rate at which mass flows in triangle through its edge denoted by symbols c . Similarly, y the second terms in the remaining two equations give the rate which momentum in x-direction,momentum in y-direction flows in triangle through its edges.The first term on right side in second and third equations gives the force in x and y direction due to pressure applied by the neighbouring elements,perpendicular to the edges,on to triangle under consideration.In most finite volume method (FVM) formulations, the flow variables are assumed to have a constant value within individual triangles but have different values in different Triangles. The assumed values are changed with time so that they are in equilibrium with their neighbouring elements; i.e. the derivatives with time (first term) in the above equations become zero.

The variables on the right hand side need to be evaluated as averaged quantities between time interval tn and tn+1. Further all these quantities are required on the cell edges. But the values are defined only in the cell. Thus these quantities must be generated based on their values given in the neighbouring cells. As a result even though the above three equations though written for an individual element, the resulting equations get coupled with the equations written for the neighbouring elements. The total number of simultaneous equations, thus will be equal to 3*riumber of elements.There are many ways of (i) averaging the variables in time and (ii) generating the values from the neighbouring cells, each of which constitute a 'numerical solution' to the above equations. These questions can not be answered here as the discussion required quite mathematical. . GRID GENERATION -In CFD, the domain is divided in to some shapes, such as triangles or quadrilaterals; and assumption is made that flows properties are constant within individual cell, but they do vary cell to cell. Obviously, the first task must be to produce these simple shapes within domain. This task is called as great generation. The set of cells are called grids or meshes. Grids must satisfy the following properties:( 1) cells must be simple in shape (say triangle or rectangle in 2D, (2) cells must completely cover the given domain,(3) no two cells must overlap and (4) cells must convex in shape. This is always true when cells are in triangular in shape, but quadrilateral need not be convex

Use of triangles as a cells very popular. Therefore procedure of triangular is referred to as grid generation. Normally a coarse mesh is first produced. This is also referred to as initial mesh. If the area of triangle is very large the triangle is divided further. The mesh generation task will continue till sufficiently small area triangles are produced .Many times instead of area, ratio of edge length are used to divide triangles to make them finer.Most of the times, triangles generated are not smooth. (1) The triangle could be elongated (2) the neighboring triangles of triangle may be either too small or toll large. Such triangulation needs smoothening.Ideally triangles should be smaller in size, where flow properties are varying rapidly. The obviously cannot in the absence of solution to the problem. But if solution is known 9say from earlier simulation), one may produce more triangles in certain region where flow properties are rapidly varying. The process is called mesh adaptation.

FINITE ELEMENT METHOD: The basic idea in FEM analysis of field problems is as follows: - The solution domain is discretized into number of small sub regions (i.e., Finite Elements).

Select an approximating function known as interpolation polynomial to represent the variation of the dependent variable over the elements.

The integration of the governing differential equation (often PDEs) with suitable Weighting Function, over each elements to produce a set of algebraic equations-one equation for each element.

The set of algebraic equations are then solved to get the approximate solution of the problem

BOUNDRY ELEMENT METHOD The boundary element method has the important distinction that only the boundary of the domain of interest requires discretisation. For example if the domain is either the interior or exterior to a sphere then the diagram shows a typical mesh; only the surface is divided into elements. Hence the computational advantages of the BEM over other methods can be considerable. The development of the BEM requires the governing PDE to be reformulated as a Fredholm integral equation .

3. NAVIER-STOKES EQUETIONS

These equations describe how the velocity, , temperature , and density of a moving fluid are related. The equations were derived independently by G.G. Stokes, in England, and M. Navier, in France, in the early 1800's. The equations are extensions of the Euler Equations and include the effects of viscosity on the flowThe equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. Recently, high speed computers have been used to solve approximations to the equations using a variety of techniques like finite difference, finite volume, finite element, and spectral methods. This area of study is called Computational Fluid Dynamics or CFD. The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass , three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t. There are six dependent variables; the pressure p, density r, and temperature T (which is contained in the energy equation through the total energy Et) and three components of the velocity ; the u component is in the x direction, the v component is in the y direction, and the w component is in the z direction, All of the dependent variables are of all four independent variables. The differential equations are therefore partial differential equations and not the ordinary differential equations that you study in a beginning calculus class. You will notice that the differential symbol is different than the usual "d /dt" or "d /dx" that you see for ordinary differential equations. The symbol look like a backwards "6", and, because of font limitations, we will use the symbol "6 /6t" on this page to indicate partial derivatives. The symbol indicates that we are to hold all of the independent variables fixed, except the variable next to symbol, when computing a derivative. The set of equations are: Continuity: 6r/6t + 6(r * u)/6x + 6(r * v)/6y + 6(r * w)/6z = 0 X - Momentum: 6(r * u)/6t + 6(r * u^2)/6x + 6(r * u * v)/6y + 6(r * u * w)/6z = - 6p/6x + 1/Re * { 6tauxx/6x + 6tauxy/6y + 6tauxz/6z} Y - Momentum: 6(r * v)/6t + 6(r * u * v)/6x + 6(r * v^2)/6y + 6(r * v * w)/6z = - 6p/6y + 1/Re * { 6tauxy/6x + 6tauyy/6y + 6tauyz/6z} Z - Momentum: 6(r * w)/6t + 6(r * u * w)/6x + 6(r * v * w)/6y + 6(r * w^2)/6z = - 6p/6z + 1/Re * { 6tauxz/6x + 6tauyz/6y + 6tauzz/6z} Energy: 6Et/6t + 6(u * Et)/6x + 6(v * Et)/6y + 6(w * Et)/6z = - 6(r * u)/6x - 6(r * v)/6y - 6(r * w)/6z - 1/(Re*Pr) * { 6qx/6x + 6qy/6y + 6qz/6z} + 1/Re * {6(u * tauxx + v * tauxy + w * tauxz)/6x + 6(u * tauxy + v * tauyy + w * tauxz)/6y + 6(u * tauxz + v * tauyz + w * tauzz)/6z} where Re is the Reynolds number 4. WIND TUNNEL TESTING v/s CFD (The CFD Advantage): -Not until the late 1960s did supercomputers begin achieving processing rates fast enough to solve the Navier-Stokes equations for some fairly straightforward cases, such as two-dimensional, slowly moving flows about an obstacle. Before then, wind tunnels were essentially the only way of testing the aerodynamics of new aircraft designs. Even today the limits of the most powerful supercomputers still make it necessary to resort to wind tunnels to verify the design for a new airplane. Although both computational fluid dynamics and wind tunnels are now used for aircraft development, continued advances in computer technology and algorithms are giving CFD a bigger share of the process. This is particularly true in the early design stages, when engineers are establishing key dimensions and other basic parameters of the aircraft. Trial and error dominate this process, and wind tunnel testing is very expensive, requiring designers to build and test each successive model. Because of the increased role of computational fluid dynamics, a typical design cycle now involves between two and four wind-tunnel tests of wing models instead of the 10 to 15 that were once the norm. Another advantage of supercomputer simulations is, ironically, their ability to simulate more realistic flight conditions. Wind-tunnel tests can be contaminated by the influence of the tunnel's walls and the structure that holds the model in place. Some of the flight vehicles of the future will fly at many times the speed of sound and under conditions too extreme for wind tunnel testing. For hypersonic aircraft (those that will fly at up to 20 times the speed of sound) and spacecraft that fly both within and beyond the atmosphere, computational fluid dynamics is the only viable tool for design. For these vehicles, which pass through the thin, uppermost levels of the atmosphere, no equilibrium air chemistry and molecular physics must be taken into account.

5. Aerodynamic analysis of formula one racing car by cfd [ TECHNIQUE OF SIMULATION ]

CASE STUDY: - Tuning various properties of the car and its appendages can modify the aerodynamic forces exerted on a Formula 1 racing car. The major goal is to provide maximum down force to facilitate power transfer from the engine, and to enhance stability especially when cornering. The front wing both provides down force and conditions the flow through the underbody, diffuser and radiator air intakes. In order to optimize the performance of the car, it is important to determine how the aerodynamic forces vary with the tuning of various parameters such as road height, wing configuration and flap angles.

GEOMETRIES AND FLOW CONDITIONS: - A number of different front-end configurations have been considered in this study. These configurations have been defined at Sauber Petronas Engineering AG. The present paper will concentrate on one particular configuration, which is shown in Fig. 1. This configuration consists of a front wing assembly, a nose section and two connecting pylons, and a ground plane. In addition, the two front wheels are included to account for flow blockage effects. To avoid large flow perturbation associated with an abrupt trailing edge, a reversed section is attached at the back of the nose section. The wing assembly consists of a main plane and, for this configuration, one flap. The wing is terminated laterally in side plates that have recessed sections near the trailing edge. The configuration is symmetric with respect to the central vertical plane (y=0).

FIG.1. FRONT- END CONFIGURATION.Two different free stream conditions were considered, 35 m/s and 70 m/s. For all computations the flow was considered to be fully turbulent. No-slip boundary conditions were applied to the wing and nose sections, i.e. the velocity on these surfaces was set to zero. The velocity of the ground plane was set to the free stream velocity. On the surface of the wheels, the tangential velocity was set to the free stream velocity, to simulate the rotation of the wheels. The flow was assumed to be symmetric with respect to the central vertical plane (y = 0), and thus only half of the flow domain was computed, reducing significantly the computational requirements. NUMERICAL TOOLS: -Numerical simulation is comprised of four main phases:GEOMETRIC MODELLING: -The configurations employed in this study have been designed using the CATIA CAD system at Sauber Petronas Engineering AG, and correspond to models that have been tested in the wind tunnel at SF-Emmen. The geometrical description of the configurations was supplied to the LMF in standard IGES format. Only slight modifications of these descriptions were found necessary for use in the numerical simulation procedure.

FIG.2: SURFACE MESH ON HALF OF FRONT-END CONFIGURATION.

MESH GENERATION: -

The generation of suitable computational meshes was performed in two stages: triangulation of all surfaces, followed by the creation of a tetrahedral mesh in the volume bounded by these surfaces. The surface mesh creation is by far the most complex and time-consuming; while surface mesh generation using cube is very flexible, it is also very user-intensive. While it is possible to refine the mesh in selected regions (e.g. boundary layers). To aid in the surface mesh generation, the geometry is divided into separate parts (e.g. wing main plane, flap, side plate, body, wheels, ground plane, etc.) and the individual surfaces meshed separately. These are then assembled to form the complete surface mesh for the computational domain. This has the advantage those modifications to the geometry (e.g. changes in the wing, ride height and wheel positions) can be accommodated without a major investment of effort.

FIG.3: MESH GENERATION.The computational domain for the present application was chosen to be a box of sufficiently large extent such that the position of the external free stream boundaries has essentially no influence on the computed flow. Surface mesh cells were concentrated in regions considered to be of most importance, e.g. on the wing. Surfaces of lesser interest, e.g. the wheels, have been allocated less mesh cells, leading to a lower accuracy in the flow solution in neighboring regions. From the computed flow solutions, it is possible to determine that there is sufficient concentration of mesh cells in the boundary layer regions. For the turbulence models used in the present study, while the number of mesh cells was adequate in the wing regions, the mesh density on the body and, in particular, on the wheels appeared insufficient.

FLOW COMPUTATION:-This flow solver uses a cell-centered finite volume discretization on an unstructured mesh, with a simple algorithm to couple the pressure and velocity fields. To obtain maximum numerical accuracy, the second-order upwind scheme was applied for the convection terms; with the diffusion terms discretized using a central difference approximation. The discretized equations were resolved using a Gauss-Seidel method coupled with a multigrid acceleration technique.

FLOW RESULT: -

Numerical flow simulations produce a wealth of data, which can be represented in a number of different manners. Only a selection of these representations is presented here. They can be divided into two classes, Local Flow fields that provide specific insights into the behavior of the flow, and Global Forces that indicate the overall aerodynamic properties.

LOCAL FLOWFIELD: - A perspective view of the contours of static pressure computed on the surfaces of the front-end section of the Formula 1 racing car is shown in Fig. 3. These views show localized regions of high pressure at the leading edges of the nose, wing, pylons and wheels. Pressure depression is computed, as expected, on the lower surface of the wings and side and rear surfaces of the wheels.

FIG.4: STREAMLINES AROUND FRONT-END CONFIGURATION. Figure 4 shows that the streamlines pass essentially undeflected laterally around and downstream of the wing, although the wheels provide a major disturbance to the flow. The trailing vortices that are generated at the side plates are not clearly apparent due to the close proximity of the wheels, resulting in a complex large-scale vortex system observed in the flow region downstream of the wing. Directly under the body, only a slight lateral displacement of the streamlines is observed. Finally, locating the wheels inboard was found not to result in major global modifications of the flow fields, as discerned from the streamline plots. Nevertheless, when the wheels are almost centrally located behind the side plates, they appear to cause an increased damping of the wing trailing vortices.

6. APPLICATIONS: - Lift and drag of aircraft. Here, as we have said, engineers need the data for performance prediction. CFD is used in conjunction with wind tunnel tests to determine the performance of various configurations. Flows over missiles. This, again, is an area where there is a need for lift, drag and side force data, so that simulations of performance can be made. As with aircraft, CFD and wind tunnel tests are used, but because of the wide range of flows that have to be simulated for a given configuration, use is also made of semi-empirical methods, which are derived from large amounts of experimental data. Jet flows inside nuclear reactor halls. Such problems involve the simulation of fault conditions, and so engineers have great difficulty in performing actual experiments, for safety reasons. Hence, computation is the only way of trying to understand such flows. Flames in burners. There is a need to understand the complex interactions between fluid flow and chemical reaction in flames. This can assist in the production of more efficient designs for burners in boilers, furnaces and other heating devices. Air flow inside internal combustion engines. When air is used to burn fuel inside an internal combustion engine, be it a gas turbine engine, a petrol engine or a diesel engine, the air must be drawn into the chamber with the minimum amount of effort and the flow of the air once it is in the chamber must be able to promote good burning. Hence, engineers need to know the pressure drop through a system and the velocity distribution in the combustion chamber.

7. LIMITATIONS

CFD-based predictions are never 100%-reliable, because: 1. The input data may involve too much guess work or imprecision.1. The available computer power may be too small for higher accuracy.2. The scientific knowledge base may be inadequate.

8. CONCLUSION Within a few years, it is to be expected, surgeons will conduct operations, which may affect the flow of fluids within the human body (blood, urine, air, the fluid within the brain) only after their probable effects have been predicted by CFD methods. Another recent and exciting application of direct numerical simulation is in the development of turbulence-control strategies that are active (as opposed to such passive strategies as riblets). With these techniques the surface of, say, a wing would be moved slightly in response to fluctuations in the turbulence of the fluid flowing over it. The wing's surface would be built with composites having millions of embedded sensors and actuators that respond to fluctuations in the fluid's pressure and speed in such a way as to control the small eddies that cause the turbulence drag.

REFERENCE

1.COMPUTATIONAL FLUID DYNAMICS By JOHN. D. ANDERSON. JR.2.www.tn.utwente.nl.com3.www.Sali.free servers.com4.www math wold.wolfram.com5.www.vision engineers.com6.www.mifu-berlin.de.com7.www-users.informatik.com8.www.boundry element method.co.uk.com

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