CFD Lab Manual

CFD LAB MANUAL

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List of Experiments: 1)Numerical Solutions of Parabolic Equation using FDM. 2)Circular Grid Generation. 3)

4)

Supersonic Flow over an infinite wing using ANSYS-CFX. Subsonic Flow over an airfoil using ANSYSFLOTRAN.





NUMERICAL SOLUTIONS OF PARABOLIC EQUATION USING FDM Let us consid consider er a parabol parabolic ic equati equation on in two indepe independent ndent variable variabless x and y. The xy plane plane is sketched in Fig, as shown

Consider a given point P in this plane. Since we are dealing with a parabolic equation, there is only only one charac character terist istic ic direct direction ion throug through h point point P. Further Furthermor more, e, in Fig. Fig. assume assume that that initia initiall conditions are given along the line ac and that boundary conditions are known along curves ab and cd. The characteristic direction is given by a vertical line through P. Then, information at P influences the entire region on one on e side of the vertical characteristic and contained within the two boundaries; i.e., if we jab P with a needle, the effect of this jab is felt throughout the shaded region shown in Fig. Parabolic equations lend themselves to marching solutions. Starting with the initial data line ac, the solution between the boundaries cd and ab is obtained by marching in the general x direction. The following types of flow-field models are governed by parabolic equations: 1) STEADY STEADY BOUNDA BOUNDAR RY-LAYER -LAYER FLOWS FLOWS





Here we will also present some of the basic aspects aspects of discretizat discretization, ion, i.e., how to replace the partial derivatives (or integrals) in the governing equations of motion with discrete numbers. Purpose of Discretization: Analytical solutions of partial differential equations involve closed-form expressions which give the variat variation ion of the dependen dependentt variab variables les contin continuous uously ly through throughout out the domain domain.. In contra contrast, st, numerical solutions can give answers at only discrete points in the domain, called grid points.

Most partial differential equations involve a number of partial derivative terms. When all the parti partial al deriva derivativ tives es in a given given partia partiall differ different ential ial equati equation on are replac replaced ed by finite finite-di -diff ffere erence nce quotients, quotients, the resulting resulting algebraic algebraic equation equation is called a difference difference equation, equation, which is an algebraic representation of the partial differential equation. Let us discretize the time & x derivative in above Eq. with a forward & central difference pattern,

After substituting the above finite difference quotients, we have

The above difference equation has two independent variables, x and t. We consider the grid as





Here, ‘i’ is the running index in the ‘x’ direction and ‘n’ is the running index in the ‘t’ direction. When one of the independent variables in a partial differential equation is a marching variable, such as ‘t’, it is conventional in CFD to denote the running index for this marching variable by n and to display this index as a superscript supe rscript in the finite-difference quotient. PROGRAM: #include #include #include float T[100][100],i,n T[100][100],i,n; ; int x,t; void display() { printf("\t\tTemperature printf("\t\tTem perature difference across the grid is:\n\n\4"); for(n=t-1;n>=0;n--) { for(i=0;i<=x-1;i++) printf("%-8.2f",T[n][i]); printf("\n\4"); }





printf("\n\n\t\t\tENTERED printf("\n\n\t\ t\tENTERED \"WRONG DATA\",\n\t\t\t DATA\",\n\t\t\tPROGRAM PROGRAM IS TERMINATING."); delay(2000); exit(); } void main() { static float r=0.3,rbt,lbt,tem r=0.3,rbt,lbt,temp; p; clrscr(); printf("MAX x and t is 100.\n"); printf("MIN x is 3 and t is 2.\n"); printf("Enter values for,\n"); printf("x="); scanf("%d",&x); printf("t="); scanf("%d",&t); if(x<3 || t<2) xit(); printf("TEMPERATURE printf("TEMPERA TURE along LEFT & RIGHT BOUNDARY:\n"); scanf("%f%f",&lbt,&rbt); printf("TEMPERATURE printf("TEMPERA TURE at (n)th LEVEL i.e. at Intial Data Line:\n"); scanf("%f",&temp); for(n=0,i=0;n<=t-1;n++) T[n][i]=lbt; for(n=0,i=x-1;n<=t-1;n++) T[n][i]=rbt; for(n=0,i=1;i






display(); getch(); }

CIRCULAR GRID GENERATION GENERATION Finite-difference approach requires the calculations to be made over a collection of discrete grid points. The arrangement of these discrete points throughout the flow field is simply called a grid. The determination of a proper grid for the flow over or though a given geometric shape is a serious matter. The way that such a grid is determined is called grid generation. The matter of grid generation is a significant consideration in CFD; the type of grid you choose for a given problem can make or break the numerical solution. The generation of an appropriate grid or mesh is one thing; the solution of the governing





int r,i; float x,y,t; clrscr(); pf("MAX RADIUS=20\nMIN RADIUS=0\n"); for(r=0;r<=20;r++) { getch(); pf("RADIUS=%d\n",r); for(i=1;i<=10;i++) { t=i*12; x=r*cos(t); y=r*sin(t); pf("%4.3f\t%4.3f\n",x,y); } } getch(); }

CFD Lab Manual

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