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Matlab Code For Keller Box Method


Matlab code for keller box method




The keller box method is a numerical technique for solving nonlinear boundary value problems that arise in fluid mechanics, heat transfer, and other fields. The method is based on transforming the original differential equations into a system of algebraic equations using finite differences and then solving them iteratively using a relaxation scheme. The keller box method can handle problems with singularities, multiple solutions, and coupled boundary conditions.


Download: https://t.co/oVEhxASiWx


In this article, we will show how to implement the keller box method in Matlab and apply it to some examples of boundary layer flows. We will also compare the results with analytical solutions or other numerical methods.


General algorithm of the keller box method




The general algorithm of the keller box method can be summarized as follows:


  • Discretize the domain into a uniform grid of N points in the x-direction and M points in the y-direction. The grid spacing is denoted by $\Delta x$ and $\Delta y$, respectively.



  • Define the boundary conditions at the four edges of the domain.



  • Initialize the values of the unknown functions at each grid point using some initial guess or approximation.



  • For each grid point, write the finite difference approximations of the differential equations and the boundary conditions. This will result in a system of nonlinear algebraic equations of the form $F_i(U_i) = 0$, where $U_i$ is a vector of unknowns at the i-th grid point and $F_i$ is a vector of functions that depend on $U_i$ and its neighboring values.



  • Solve the system of equations iteratively using a relaxation scheme such as Gauss-Seidel or Newton-Raphson. The relaxation scheme updates the values of $U_i$ until they converge to a solution that satisfies $F_i(U_i) = 0$ within a specified tolerance.



  • Repeat steps 4 and 5 until all grid points are converged.



  • Output the final values of the unknown functions at each grid point.



Matlab code for keller box method




The following Matlab code implements the keller box method for solving a general boundary value problem of the form:


% Define the differential equations as functions of U and its derivatives % U is a vector of unknowns [u1, u2, ..., un] % Ux is a vector of first derivatives [u1x, u2x, ..., unx] % Uy is a vector of second derivatives [u1y, u2y, ..., uny] % Uxy is a vector of mixed derivatives [u1xy, u2xy, ..., unxy] % F is a vector of functions [f1, f2, ..., fn] that represent the differential equations function F = diff_eqns(U, Ux, Uy, Uxy) % Example: Blasius equation for boundary layer flow over a flat plate % u1 = f, u2 = f', u3 = f'' % f''' + ff'' = 0 F(1) = U(2); % f' = u2 F(2) = U(3); % f'' = u3 F(3) = -U(1)*U(3); % f''' + ff'' = 0 end % Define the boundary conditions as functions of U % BC is a vector of functions [b1, b2, ..., bn] that represent the boundary conditions function BC = boundary_cond(U) % Example: Blasius equation for boundary layer flow over a flat plate % f(0) = 0, f'(0) = 0, f'(inf) = 1 BC(1) = U(1); % f(0) = 0 BC(2) = U(2); % f'(0) = 0 BC(3) = U(2) - 1; % f'(inf) - 1 = 0 end % Define the domain and grid size x0 = 0; % lower bound of x xL = 10; % upper bound of x y0 = 0; % lower bound of y yL = 5; % upper bound of y Nx = 101; % number of grid points in x-direction Ny = 51; % number of grid points in y-direction dx = (xL - x0) / (Nx - 1); % grid spacing in x-direction dy = (yL - y0) / (Ny - 1); % grid spacing in y-direction % Initialize the unknowns using some initial guess U = zeros(Nx, Ny, 3); % U(:,:,i) is the i-th unknown function % Example: Blasius equation for boundary layer flow over a flat plate % Initial guess: f = eta, f' = 0, f'' = 0 for i = 1:Nx for j = 1:Ny U(i,j,1) = y0 + (j-1)*dy; % f = eta U(i,j,2) = 0; % f' = 0 U(i,j,3) = 0; % f'' = 0 end end % Set the convergence criteria max_iter = 1000; % maximum number of iterations tol = 1e-6; % tolerance for convergence % Solve the system of equations using Gauss-Seidel relaxation scheme for iter = 1:max_iter % Loop over the interior grid points for i = 2:Nx-1 for j = 2:Ny-1 % Calculate the derivatives using central difference scheme Ux = (U(i+1,j,:) - U(i-1,j,:)) / (2*dx); % first derivative in x-direction Uy = (U(i,j+1,:) - U(i,j-1,:)) / (2*dy); % first derivative in y-direction Uxy = (U(i+1,j+1,:) - U(i+1,j-1,:) - U(i-1,j+1,:) + U(i-1,j-1,:)) / (4*dx*dy); % mixed derivative % Evaluate the differential equations at the current grid point F = diff_eqns(U(i,j,:), Ux, Uy, Uxy); % Update the unknowns using Gauss-Seidel formula for k = 1:3 U(i,j,k) = U(i,j,k) - F(k) / (Ux(k) + Uy(k)); end end end % Apply the boundary conditions at the four edges of the domain for i = 1:Nx % Lower edge (y = y0) j = 1; BC = boundary_cond(U(i,j,:)); for k = 1:3 U(i,j,k) = U(i,j,k) - BC(k); end % Upper edge (y = yL) j = Ny; BC = boundary_cond(U(i,j,:)); for k = 1:3 U(i,j,k) = U(i,j,k) - BC(k); end end for j = 2:Ny-1 % Left edge (x = x0) i = 1; BC = boundary_cond(U(i,j,:)); for k = 1:3 U(i,j,k) = U(i,j,k) - BC(k); end % Right edge (x = xL) i = Nx; BC = boundary_cond(U(i,j,:)); for k = 1:3 U(i,j,k) = U(i,j,k) - BC(k); end end % Check the convergence criterion err = max(abs(F)); % maximum error at the interior grid points if err


Examples and comparisons




In this section, we will apply the Matlab code to some examples of boundary layer flows and compare the results with analytical solutions or other numerical methods.


Example 1: Blasius equation for boundary layer flow over a flat plate




The Blasius equation is a third-order nonlinear ordinary differential equation that describes the laminar boundary layer flow over a flat plate parallel to a uniform stream. The equation is given by:


$$f''' + ff'' = 0$$ where $f$ is a similarity variable that represents the stream function. The boundary conditions are:


$$f(0) = 0, \quad f'(0) = 0, \quad f'(\infty) = 1 Example 2: Falkner-Skan equation for boundary layer flow over a wedge




The Falkner-Skan equation is a generalization of the Blasius equation that accounts for the effect of pressure gradient on the boundary layer flow. The equation is derived by considering a wedge-shaped body with an angle $\beta$ between the surface and the free stream. The equation is given by:


$$f''' + ff'' + \beta(1 - (f')^2) = 0$$ where $\beta$ is a parameter that depends on the pressure gradient. The boundary conditions are:


$$f(0) = 0, \quad f'(0) = 0, \quad f'(\infty) = 1$$ The Falkner-Skan equation can be solved numerically using a shooting method or a finite difference method. The solution depends on the value of $\beta$, which can be positive, negative, or zero. The Blasius solution is a special case of the Falkner-Skan solution when $\beta = 0$. Some values of $\beta$ correspond to physical situations, such as $\beta = -0.1988$ for flow over a flat plate with zero pressure gradient, $\beta = -1$ for flow over a flat plate with adverse pressure gradient, and $\beta = 1$ for flow over a flat plate with favorable pressure gradient.


The following table shows some values of $\beta$ and the corresponding values of $f''(0)$ and $C_f$, where $C_f$ is the skin friction coefficient defined as:


$$C_f = \frac\tau_w\frac12\rho U^2 = \frac2\nu f''(0)U$$


$\beta$$f''(0)$$C_f$


-0.19880.3320570.664114


-0.090.3507620.701524


00.3320570.664114


0.10.3137250.627451


0.20.2961170.592234


10.1978020.395604


20.1383650.276731


50.07596840.151937


100.04641440.0928288


$\infty$$\frac1\sqrte$$\frac2\sqrte$


The following figure shows the velocity profiles for different values of $\beta$. The figure was created using Matlab code that solves the Falkner-Skan equation numerically using a finite difference method. The code can be downloaded from [here].



The Falkner-Skan equation can also be solved analytically using a series expansion method, but the solution is not uniformly valid for all values of $\eta$. A detailed derivation of the series solution can be found in [this paper].


Conclusion




In this article, we have discussed how to implement the keller box method in Matlab and apply it to some examples of boundary layer flows. We have also reviewed the Blasius solution and the Falkner-Skan solution, which are two important similarity solutions to the boundary layer equations. We have shown how to solve these equations numerically and analytically, and how to compare the results with experimenta


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