History of development of CFD from point orient to

Numerical Simulations with Parent Methods. Kaleeswaran.B

Department of Aerospace engineering,

UPES, Uttrakhand

kaleesmach@gmail.com

Abstract— this paper reveals about the history of CFD from

the starting years till the present scenario. The changes CFD

had bought to the world are also discussed along with its

applications in present and future industrial needs. Many

methods like FEM, FDM were also explained in brief.

Keywords— CFD, FEM, FDM.

I. INTRODUCTION

CFD helps us to understand even sophisticated ideas like

flow through veins, blood vessels etc.CFD helps in shorter

numerical analysis of flows, which is an advantage over

long time wind tunnel analysis.CFD helps to design models

to understand various flow properties under various

inaccessible conditions also.The basic equation of CFD

solutions is Navier-Stokes equations which defines the

character of singular phase fluid flow. Simplification of NS

equation leads to Euler’s equation which describes vorticity

further simplification leads to full potential equation which

determines vorticity and further simplification of it gives

linearized potential equations.

TABLE I: Historical development of CFD

s.no YEAR DEVELOPMENTS

1. 1928 Numerical solution method for PDE

S was developed.

Parabolic time marching method.

2. 1943 Improvement of RITZ method.

3. 1950 Explicit forward time difference

method was developed.

4. 1950-1960 In viscid methods were developed.

5. 1960 FEM was developed for engineering

models.

6. 1968 Artificial compressibility effect[2]

7. 1970 Numerical solution method.

8. 1970-1980 Improvement of FEM for arbitrary [6]

geometry.

9. 1970 For fluid study FDM was developed.

Relaxation methods.

10. 1990 DNS method.

11. 20 th

century

Closed form of numerical

solutions[1], analytical geometries.

II. Historical applications of cfd:

Early applications:

1.1960 - NASA, BOEING developed panel methods to

design aircraft models, vorticity stream function analysis

methodology.

2.1970 - metrological survey for turbulence[7]

measurement

of cyclones with large eddy simulation models, SIMPLE

algorithm (a parabolic flow code)

3.1980- industrial applications in CFD like CAD, FEA

were developed.

Reference: http://www.cfd-online.com

III. Methodology used in CFD:

1. Geometry of the flow is first defined.

2. The volume occupied is then divided into discrete cells.

3. The physical modelling is then defined.

4. The boundary conditions are defined at last.

5. The simulation is the next process followed by

postprocessor results.

Discretization is defined as the process of splitting volumes

into N number of cells.

.

IV .IMPORTANT METHODS IN CFD:

IV.1.FDM (Finite Difference Method):

Commonly used method in CFD analysis is FDM.In this

method domain is specified or covered by grids [4]

.FDM

uses finite differences approximation techniques for

discretization.

Fig 1: FDM 1 D equation.

FDM has the following procedures;

1. Discretize the solution domain by a fine grid.

2. At each point finite difference equation is converted to

algebraic equation.

3. Boundary conditions and interior points are coined.

4. Summation of all algebraic equations both interior and

boundary conditions to obtain algebraic equations for

unknown values.

5. Solve algebraic equations to get variables at each point-

interpolated values were used for post processing results.

Governing equation is given by

This method uses Taylor’s series and polynomial fittings

were used. Results obtained were summed up with

truncation errors to get confined results.

IV.2.FEM (Finite Element Method)

It depends purely on WR method .It is used in analysis of

mainly solids and also applicable to fluids. It is much stable

than FVM.The stationary points in FEM [5]

are of

differential forms of conservation equations.WR method

transforms this differential form to integral form.

Weighted residual form is;

Where Ri is residual equation at element i, Q is

conservation equation, Ve is volume of element,Wi is the

weight factor.

Steps included in FEM are as follows:

1. Discretize solution domain by a mesh and choose

appropriate set of interpolation functions.

2. Weak form of WR residual governing equation is applied

and evaluates required integrals.

3. Combine all elements to form global discrete system.

4.Apply boundary conditions; solve global system of

algebraic equations to obtain value at each node.

FEM has the following solution methods;

1. Point collacation method,

2. Subdomain method.

3. Petrov-Galerkin method.

4. Boundary element method.

IV.3.FVM (Finite Volume Method)

Approximate solution of integral form of conservation

equations. The problem is divided into finite volumes.FVM

involves use of approximate integral formula and

interpolation nodes.FVM can use any type of grids .It can

mesh even complex geometry. Widely used CFD package.

It steps following steps;

1. Discretize solution by a grid. Apply integral form to it.

2. Collect algebraic forms of it for all finite volumes.

3. Solve algebraic forms to obtain variables.

Types:

Cell centred approximation.

Face centred approximation.

Figure 2: left- cell centred, right- face centred.

IV.4.TURBULANCE MODELING:

LES: They are functional in between DNS and modeling

approach. Large scale models were computed to give

appropriate results. In this method small eddies were

studied for solving the problem. It shows good results when

compared to RANS.

RANS:

Turbulence modeling involves 4 fluctuation variables like

velocity(V),pressure(P),temperature(T),and density (ῥ).

Flow variables at a given point can be represented as

average of the above four fluctuating factors which is

termed as Reynolds averaging also called as Reynolds

decomposition.

Obtaining RANS to continuity, momentum equations are

given by;

A RANS model is divided into two approaches:

1. Boussinesq hypothesis.

2. Reynolds stress model.

IV.5.NUMERICAL SIMULATION:

Numerical method changes transport equation to algebraic

equations. Numerical simulations are then converted to

numerical simulation of space and time which is termed as

descriptive form of solutions. Numerical solutions are

mainly used in heat transfer problems. [3]

History of Numerical solutions:

Semi analytical methods were used as perturbation methods.

Reduction method solves biharmonic and laplacian

methods. CFL condition is used to convert PDE to

numerical simulations.Lax and wendroff developed

MCcormark solutions to solve second order FDM.

IMPLICIT SCHEME was developed by Patankar and

Spalding using SIMPLE algorithm which was later

developed to SIMPLEC, SIMPLEX.

Role of Numerical solutions in technology environment:

Numerical analysis is an emerging subject in modern world.

It’s a combination of both analytical and experimental

solutions. It helps to predict a problem that leads to product

development. It’s of less cost and finds good application in

heat transfer problems and wind tunnel testing.

Longitudinal vortices:

It’s a flow control method. It disrupts growth of boundary

layer and enhances heat transfer. Main vortex system is

used in winglet termed as horse shoe vortex.

Petrov galerkin method:

PG method for spatial discretization method and for high

Reynolds number.

CONCLUSION

CFD is a vast subject and its journey of complete

emergence is also so vast. A few important dates of its

important steps were given and few methods were explained.

Acknowledgement:

I gratefully acknowledge Dr.Ugur Guven coordinator CFD

and Mr.Gurunath Valedi for inspiring us to make a paper

presentation on CFD aspects.

REFERENCES

[1] Van Dyke, M., Perturbation Methods in Fluid Mechanics,

Academic Press (1964).

[2] Von Mises, R., Mathematical Theory of Compressible Fluid Flow,

Academic Press (1958). [3] Patankar, S.V., and Spalding, D.B., Int. J. Heat Mass Transfer, 15,

1787-1806 (1972).

[4] Thompson, J.F., Warsi, Z.U.A. and Mastin, C.W., Numerical Grid

Generation (1985).

[5] Taylor, C. and Hughes, T.G., Finite Element Programming of the

Navier-Strokes Equations, Pineridge Press (1987).

[6] Hess, J.L.; A.M.O. Smith (1967). "Calculation of Potential Flow

about Arbitrary Bodies". Progress in Aeronautics Sciences.

[7] Wilcox, David C. (2006). Turbulence Modeling for CFD (3 ed.). DCW Industries, Inc.

[8] Anderson, John D. (1995). Computational Fluid Dynamics: The

Basics with Applications. Science/Engineering/Math. McGraw-Hill

Science.