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This is a scientific web page about the two-dimensional steady incompressible flow in a driven cavity. The steady incompressible 2-D Navier-Stokes equations are solved numerically.

Driven cavity flow, numerical methods, steady incompressible flow, finite difference, Navier Stokes equations.

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I discuss about physical, mathematical and numerical aspects of these flows, post many figures and tables, also post fortran codes, solution datas and etc. In this study Erturk et. Using this numerical method Erturk et. I post all of my numerical solution data presented in the manuscript and also the fortran solver used in this study available for download.

In this study Erturk E. Erturk also presented very very fine grid solutions of driven cavity flow simply solved with using Successive Over Relaxation Method SOR. Using this formulation, any existing solver that solves the 2-D Steady Incompressible Navier-Stokes equations with second order spatial accuracy can be easily modified to provide fourth order spatial accuracy.

With this formulation they solved the flow in a driven cavity up to Reynolds number of 20, with fourth order spatial accuracy using very fine grid mesh. Detailed solutions of flow inside various tringle cavity geometries are presented.

The numerical method proved to be very efficient for the solution of N-S Equations in general curvilinear coordinates in non-orthogonal geometries even at extreme skew angles. Detailed solutions are presented for future references.

I post all of my numerical solution data presented in the manuscript available for download. Using the efficient and stable numerical method described above in " Study 1 " the backward facing step flow is solved for high Reynolds numbers.

Detailed results are presented. Methods Fluids50 In finite difference, one can obtain fourth order O D x 4 accurate numerical solutions using the compact formulation. One can also obtain fourth order accurate O D x 4 numerical solutions using the standart five point wide discretization. In this study, the fourth order compact formulation introduced by Erturk and Gokcol Int. Methods Fluids50is extended to non-uniform grids.

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The advantage of this formulation is that the presented formulation provides fourth order O D x 4 spatial accuracy on non-uniform grids. The efficiency of the formulation is demonstrated on the lid driven cavity flow benchmark problem.

The polar cavity is considered as both driven from inside wall and from outside wall. The computed results are compared with experimental and numerical results. Detailed numerical results of the driven polar cavity flow are presented. Using a very efficient numerical method and a very large mesh, numerical solutions are obtained up to Reynolds number of It is found that the solution of 2-D steady incompressible flow around a circular cylinder change behavior around Reynolds number of and Detailed numerical results of the flow over a circular cylinder are presented.

Using an efficient finite difference numerical method together with a very large mesh, highly accurate steady solutions are obtained up to Reynolds number of Detailed numerical solutions are presented. The governing 2-D steady incompressible Navier-Stokes equations are solved with a very efficient finite difference numerical method using a very large stretched mesh such that the inflow and the outflow boundary is located very far away from the square cylinder.

Detailed results of the flow characteristics are presented. For each of the considered blockage ratio the flow past a square cylinder confined in a channel is simulated up to very high Reynolds numbers.

The numerical solutions of different channel blockage ratios are compared with each other and detailed results are presented. The solutions of the presented compact streamfunction and vorticity formulations are spatially forth order accurate.

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These compact formulations provide high spatial accuracy even with using coarser grid points in the computational domain. The efficiency of the formulation is demonstrated on the driven polar cavity flow benchmark problem.Numerical simulator for lid driven flow problem Incompressible Navier Stokes Equation.

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This repository basically combines all the data required for understanding the Lid Driven Cavity and Navier Stokes Equation's Numerical Solution by taking help from the 12 steps to Navier Stokes eq by lorena Barba. Add a description, image, and links to the lid-driven-cavity topic page so that developers can more easily learn about it.

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Reload to refresh your session. You signed out in another tab or window.The aim of this paper is to study the properties of approximations to nonlinear terms of the 2D incompressible Navier-Stokes equations in the stream function formulation time-dependent biharmonic equation. The nonlinear convective terms are numerically solved by using the method with internal iterations, compared to the ones which are solved by using explicit and implicit schemes operator splitting scheme Christov and Marinova; Using schemes and algorithms, the steady 2D incompressible flow in a lid-driven cavity is solved up to Reynolds number Re with second-order spatial accuracy.

The schemes are thoroughly validated on grids with different resolutions. The result of numerical experiments shows that the finite difference scheme with internal iterations on nonlinearity is more efficient for the high Reynolds number. There are many finite difference methods for the solution of the Navier-Stokes equations NSEs representing incompressible viscous flows. Some of these are schemes utilizing primitive variables velocity-pressure [ 1 — 4 ], vorticity-stream function [ 5 — 9 ], and stream function formulation [ 1011 ].

The practical estimation of any schemes may be different from the theoretical estimation because of the nonlinearity of the NSEs and the implicit characteristic of the continuity condition.

There is no single method which is most suitable for all aspects. The solution of the full-nonlinear set of discretized fluid flow equations is usually obtained by solving a sequence of linear equations.

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The type of linearization which is used can significantly affect the rate of convergence of the iterations to the solution. The primary difficulty in obtaining numerical solutions with primitive variable formulation is that there is no evolution equation for pressure variable. The pressure function can be removed from the equations by means of vorticity-stream function, but an explicit boundary condition for vorticity function is missing. The simplest way to avoid the case is to eliminate the vorticity and turn to a single nonlinear biharmonic equation for the stream function.

The condition of stability of an explicit scheme imposes very restrictive limitations on the time increment with the artificial time. For the fourth-order derivative, it is. That is why the problem of constructing an implicit scheme is of great significance. The straightforward implementation of an implicit scheme results in the very large linear system whose solution needs very large computer capacity. One of the most efficient approaches to reduce the computational time without compromising the stability of the scheme is the method of operator splitting [ 12 ].

The application of the splitting method to the NSEs in stream function formulation is not straightforward because of the fact that they are not a Cauchy-Kowalevska system and there is no time derivative of the stream function. What is present in the equation is the term which cannot serve as a basis for splitting [ 1 ]. The new scheme combines the computational simplicity of the implicit scheme in linear terms with semiexplicit approximation of nonlinear terms.

The general idea when treating the nonlinear term is to represent it as implicit approximation and then to linearize it and to conduct internal iterations. After the inner iterations converge, one obtains, in fact, the solution for the new time stage. The explicit approximation of nonlinear terms accomplish severe requirement on time step. A single internal iteration on nonlinear terms induces sense of implicit approximation and reduce very severe band on time step.

To improve stability properties of explicit approximation of nonlinear terms we require only 3 internal iterations and call such algorithm as a method with internal iteration. As to the question of which method to be used, the answer is that it depends on the type of problems that to be solved. The objective of the present study is to validate the efficiency and accuracy of numerical implementations for the NSEs in term of stream function.

The content of this paper is organized as follows. Section 2 contains the mathematical formulation of the problem. Section 3 deals with the discretization of the equations and detailed description of numerical algorithms. The methods are applied to the test problem of lid-driven cavity flow up to. Results and discussions of numerical solutions are presented in Section 4where we make a detailed comparison with available numerical data.

Consider a closed 2D domain with a piecewise smooth boundary. The NSEs for a viscous incompressible flow in the terms of stream function are where. Boundary and initial conditions are the following: where is a vector normal to domain boundary, and the Reynolds number is defined aswhere is the characteristic velocity, is the characteristic length, and is the kinematic viscosity.

The lid-driven flow occupies the region the cavity No-slip boundary conditions take the following form: For the sake of simplicity, we assume a uniform grid with and spacing in and -direction, respectively, for and. The mesh is staggered in direction on and in direction on with respect domain boundaries. The boundary conditions are approximated on two-point stencils with the second order of approximation as follows: where with.Hello everyone Lid driven cavity problem is a very well known problem and has been solved many times in the past.

In this problem a fluid is contained in a rectangular box with three fixed sides and moving top part lid. Then the lid is given some velocity due to which the fluid in direct contact with the lid also starts moving and then transfer the motion to next layer below.

Then continuing in this fashion the motion is transferred in the whole fluid. The sense of motion in the fluid depends on the velocity of the lid. At different velocities different patterns of motion is formed in the box. Therefore time will not be wasted in debugging the side subroutines but the focus will be more on the main process. The code will smaller in lines to write and will be easy to understand. Different files have been created for every function used in the code.

The folder including every file has been attached with this post. Now as you know about the problem I will discuss about the process of solving this problem.

Now, to remove the pressure oscillations that may happen the staggered grid arrangement will be used as shown below. Now, with the use of the intermediate velocities calculated the pressure poisson equation will be solved to calculate the pressure values. The pressure poisson equation is given below. Discretization is done for the terms in above mentioned pressure poisson equation.

Now when this above discretized equation will be solved for each node in the grid then it will result into a matrix equation of the form. Boundary conditions for pressure are applied according to the figure given below. Now, the third step i. When the above equation is discretized for u and v velocities, we have. Hence the final velocities will be calculated. Now plotting the contours of velocity, pressure and centreline velocity is an easy job which can be calculated using the values calculated by the above mentioned process.

But, the contours of Stream function and Vorticity is explained below. For Stream function contours the following equation is used. For vorticity contours the following equation is used.

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I have entered relevant comments in the code itself so that you can relate the process with the code step by step. Appropriate description is written in the comments so that step by step working can be known. First you should understand the file named navierStokes. This will maintain the rhythm and sequence of the program and you will be able to understand the program wisely.

The main code and the other function files are very easy to understand as there are one line inbuilt functions for very complex C codes. Therefore it is easy to code in comparison to C language. So, now you know the concept of the method and working of the code.Updated 08 Aug Staggered grid for u and v. To see how indexing works in staggered grid, Please check out "Versteeg, Malalasekera: an introduction to computational fluid dynamics" text book.

Discretization of the governing equations is based on this text book. Pressure correction equation is directly solved using a penta-diagonal matrix algorithm in every iteration.

Proper choice of under-relaxation factors needed for convergence. Jacobi method is the least efficient way for this type of problems but it's simple and easy to prallelize. So, clamp the pressure of one node in the domain to zero as a boundary condition and the pressure at other nodes will be measured relative to that point. Other boundary P' values will be set this way. I am going to upload it on GitHub pretty soon. It's much faster than this version because: 1 It is parallel and not serial so you can you can use multiple processors.

MJ Sarfi Retrieved April 11, Can someone tell me why the velocities in the momentum equations are evaluated explicitly? Shouldn't they be calculated using neighbouring velocities at the same time step?

Great work! I would appreciate any help. Fixed a mistake in assembling the matrix of coefficients for the pressure correction equation Fixed another mistake about the defining the Peclet number in power-law upwinding scheme. Learn About Live Editor. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance.

Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation. File Exchange. Search MathWorks. Open Mobile Search. Trial software. You are now following this Submission You will see updates in your activity feed You may receive emails, depending on your notification preferences.

Follow Download. Overview Functions. Cite As MJ Sarfi Comments and Ratings Deep Morzaria Deep Morzaria view profile. Mingming Zhang Mingming Zhang view profile. Mohammad Mohammad view profile. Robin Robin view profile. Yerkanat aydarken Yerkanat aydarken view profile. Updates 8 Aug 1.Writing your own solvers is fun, rewarding, and is a practice that really cements some of the fundamental knowledge of CFD.

The cavity flow problem is described in the following figure. Basically, there is a constant velocity across the top of the cavity which creates a circulating flow inside. At high Reynolds numbers we expect to see a more interesting result with secondary circulation zones forming in the corners of the cavity.

The two equations above are easily derived and I will probably make a short blog post covering it. For now you can take my word for it… these solve a major complication of the Navier-Stokes equations and that is the pressure-velocity coupling! By making a change of variables from u, v, and p, to and this is now in a simpler form to solve numerically. Quickly I need to explain the boundary conditions for this problem.

They seem simple enough, velocity top and no velocity on the left, bottom and right sides. But remember there is no velocity in our equations! We have stream-function and vorticity. So how do we set our boundary conditions?!

Well, the trick is to recognize that the outside can be considered a closed streamline.

### Numerical Implementations for 2D Lid-Driven Cavity Flow in Stream Function Formulation

Meaning that all around the edge it will have a constant value of stream function. What should the value be? And what about vorticity? It is an output of the simulation. At each iteration of the solver we must calculate the value of the vorticity at the wall either from the stream function or by using the same method for calculating the rest of the vorticity points.

Now, we are going to use finite differences to discretize the equations and create a system of equations that can be solved. Finite differences are used to approximate the derivatives at each point using the surrounding points that are a finite distance away.

This is shown in the next figure. By doing this we create one equation for each node inside the cavity. When all of these equations are put together they form a system of equations that can be solved.Updated 10 Sep Cavity flow is simulated using the pressure correction method on a staggered grid using explicit differencing for the hyperbolic terms CD, MacCormack and Richtmyer method while both explicit and implicit methods are considered for the diffusive parabolic terms.

For the implicit steps, preconditioned matrices are used using LU decomposition. The Pressure Poisson equation is also solved implicitly. Neumann boundary conditions are used for pressure and Dirichlet conditions for the velocity field. Suraj Shankar Retrieved April 11, The result seems incorrect, due to a strange recirculation in the upper right corner.

Lid Driven Cavity Flow using MATLAB 2014A

The code is meant to be pedagogical in nature and has been made in line with the steps to Navier-Stokes practical module, for which I would like to credit Lorena Barba and her online course on CFD. Learn About Live Editor. Choose a web site to get translated content where available and see local events and offers.

## Solving the cavity flow problem using the streamfunction-vorticity formulation

Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation. File Exchange. Search MathWorks. Open Mobile Search. Trial software. You are now following this Submission You will see updates in your activity feed You may receive emails, depending on your notification preferences. Follow Download. Overview Functions. Cite As Suraj Shankar Comments and Ratings Luiz Miranda Luiz Miranda view profile.

Radu Trimbitas Radu Trimbitas view profile. Sagar Sagar view profile. Shubham Maurya Shubham Maurya view profile. Seems to be caused by the projection step.