Source, sink,doublet, vortex, spiral flow, flow past a cylinder with circulation-Magnus effect-Joukowski theorem. 4 s along lines passing through the vortex core center while parallel to r and z axis. This MATLAB function computes the curl (curlx, curly, curlz) and angular velocity (cav) perpendicular to the flow (in radians per time unit) of a 3-D vector field U, V, W. Laplacian flow의 의미는? 8. Note that it is constant for the constant rotation V = Arand zero. 2 Stream Function Formulation (Using Variational Approach) 592 19. عرض ملف Navheen Shanmugham Murugesan الشخصي على LinkedIn، أكبر شبكة للمحترفين في العالم. Lax-Wendroff, MacCormack’s methods 13 Numerical solution of unsteady compressible Navier-Stokes and Energy equations. This routine might be costly since it inverses the Laplacian of the vorticity (using a successive over relaxation solver). The average velocity field of Hele-Shaw flow is an physical analog for the irrotational, solenoidal, 2-D flow described by the complex potential. Hence, according to Section 5. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. -Developed a 2D Computational Fluid Dynamics(CFD) Solver program for a backward facing step flow problem by implementing vorticity stream-function approach using FORTRAN -Developed MATLAB programs for correctly plotting the results generated by the previous FORTRAN programs -Performed grid dependence studies on all the above. Description. Title: Microsoft Word - IRJABS-13-546 Author: M & R Created Date: 7/16/2013 2:21:01 PM. Refer to Matlab's help page for curl function, it's a good reference for both computing vorticity and slicing a. The spatial behavior. 8 governing equations in cavity flow 17 2. * MATLAB 5 (MATLAB R13), * Basic (Visual Basic 6/7 3). The surrogate function interpolates the critical eigenvalues obtained from. Defining stream function and velocity potential function. Moreover two-dimensional potential flows can be constructed from any analytic function of a complex variable, W(z). Solve the lid driven cavity flow using vorticity-stream function formulation. Boffetta and G. 5 and t = 1 s, and for steady state (steady state is reached when, for two successive time instances t n + 1 and t n for both vorticity and stream function field values, N R M S E / d t < 10 − 8). Mathematical tools such as the potential function, stream function, and complex potential are concisely outlined in Appendix A. Ducted turbulent flows with varying wall shapes are formulated with the Navier-Stokes equations written in terms of vorticity and stream function. Stream function; the magnetization direction perpendicular to the flow direction Fig. Kammalapati, Propagation of Shock on NREL Phase VI Wind Turbine Airfoil under Compressible Flow , Journal of Renewable Energy, vol. Then what we get is a discretized stream function $\,\psi(x,y)$. -Developed a 2D Computational Fluid Dynamics(CFD) Solver program for a backward facing step flow problem by implementing vorticity stream-function approach using FORTRAN -Developed MATLAB programs for correctly plotting the results generated by the previous FORTRAN programs -Performed grid dependence studies on all the above. The elements of A and B are input into the program following the basic syntax of MATLAB programming. Two lasers were necessary in order to avoid shadows when measuring velocities in flows around a solid body. ENGINEERING MATHEMATICS (M-101) Higher engineering mathematics by B. 3 Helmholtz’s Theorems 159 10. Incompressible fluid flow, stream function-vorticity formulation. Dubois F, Salaün M, Salmon S. MS = 'MarkerSize'; 17. The complex velocity is independent of the path along which the derivative is of the. ; L=10; M=10; N=1; LX=1; LY=1; DX=LX/L; DY=LY/M; for i=1:L for j=1:M; for k=1:N. here centered on timestep n. 9 Conclusions. Learn more about streamfunction, poisson equation, vorticity, velocity field. Fluid Dynamics, Stream function in polars. Kammalapati, Propagation of Shock on NREL Phase VI Wind Turbine Airfoil under Compressible Flow , Journal of Renewable Energy, vol. I thus calculate the vorticity $\omega$ (such that $\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$) and then solve the Poisson equation $abla^2 \psi=\omega$. 2 The Operations *, / , and ^. Solve for the stream function! Find vorticity on boundary! Find RHS of vorticity equation! Initial vorticity given! t=t+!t! Update vorticity in interior! Solution Strategy! Computational Fluid Dynamics! for i=1:MaxIterations! for i=2:nx-1; for j=2:ny-1! s(i,j)=SOR for the stream function! end; end!. pdf: refernce module 4: 10: Solution of Navier-Stokes Equations for Incompressible Flow Using SIMPLE and MAC Algorithms: reference_mod5. The shape of the velocity curve (the velocity profile across any given section of the pipe) depends upon whether the flow islaminar or turbulent. 9 Bibliography. Matlab code for fluid flow. This approximation reduces the problem to a set of matrix equations whose term involving derivatives of time is the mass matrix. Magnetic field, B and Stream function ψ taking them as function of x, y and. employing the stream and vortices functions [7]. In this article, we first develop a semi-discretized Crank–Nicolson format about time for the two-dimensional non-stationary Stokes equations about vorticity–stream functions and analyze the existence, uniqueness, stability, and convergence of the semi-discretized Crank–Nicolson solutions. MS = 'MarkerSize'; 17. The ability to discover physical laws and governing equations from data is one of humankind’s greatest intellectual achievements. 2) and ∇2ψ= ψ00+ψ0/r−m2ψ/r2. , Tracey et al. Fluid Flow Velocity Profiles. It is also a physical analog for 2-D flow of incompressible fluids through porous media by. We deﬁne the vorticity to be the curl of the two-dimensional ﬂow, where [11] ω= curl~v= ∂v. After computing initial values for the vorticity field, the iteration starts with solving for the streamfunction using the Jacobi Iteraition. I have position data for x and y, as well as velocity data in the x and y directions. Chapter 4 Part B: Stream function, Vorticity, Cauchy-Riemann, Boundary layer, Approximate solution - Materials are with copyright. m (Solves the velocity function) 4. Fluid Dynamics, Stream function in polars. Vorticity 2d - Cfd - Matlab. Filter by. Goudnov solution of shallow water equations and stream function-vorticity formulation for two-dimensional. 5 and t = 1 s, and for steady state (steady state is reached when, for two successive time instances t n + 1 and t n for both vorticity and stream function field values, N R M S E / d t < 10 − 8). Navier-Stokes equations are solved and velocity and pressure are calculated Jun 22, 2018 · Kindly send the matlab code for load flow analysis [email protected] NAST2Dis a FORTRAN90 program which implements the finite volume method to solve for the transient velocity, pressure, and temperature of an incompressible fluid in a variety of 2D flow regions. only the -component can be non-zero. Then we establish a fully discretized Crank–Nicolson finite spectral element format based on the. stream function (unit: Sv, 1 Sv = 106 m3 sec 1) with the absolute geostrophic velocity for the extra-equa-torial region (north of 8°N and south of 8°S) calculated by the P-vector method (Chu and Fan, 2007) as exam-ples. s for the stream function is quite simple from its definition in terms of the velocity field. other GUI is a MATLAB box, that requires input from the user for the speed of the uniform flow over the airfoil, the airfoil angle of attack, and the vorticity of the flow field, due to the object rotating in the flow field. Stream functions: A stream function, Ψ, is the function used to plot streamlines and tell us the speed of the flow/the volume flow rate. The stream function and vorticity equations can be solved using the finite difference method. Differences in LEV vortex strength as a function of chord length will be presented. 6 The Complex Velocity 35 2. The reactor is equipped with two disk impellers. Particle in Cell Consulting LLC is a small California-based company specializing in the development of high performance computer simulation programs for electric propulsion plasma thrusters, plume-spacecraft interactions, and molecular and particulate contamination transport. m (Implements the upwind scheme) 7. The spatial derivatives of vorticity and the stream function must be computed each time-step. its value only depends on the locations of the points A and P. Fractal Metaphor 3. See full list on curiosityfluids. Differences in LEV vortex strength as a function of chord length will be presented. The calculations are made by a computer program, written in MATLAB. A streamline is a path traced out by a massless particle as it moves with the flow. Flow Net – Orthogonality of stream lines and equipotential lines. X, Y, and Z must have the same number of elements, as if produced by meshgrid. Results, details of the numerical algorithms, as well as numerical verification techniques and validation with sources from the literature will be presented. Calculate the vorticity eld. 1 Introduction. 3 Defining and Plotting Functions in MATLAB. The stream function equation is discretized using the standard central difference, and can be solved using an iterative elliptic solver, such as Jacobi or Gauss-Seidel. See also Russell and Wang (2003) for an alternative method employing Cartitian grids appropriate for multiple wings. X, Y, and Z must have the same number of elements, as if. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part. The results were in a very good agreement. NavierStokes. The equations concern the vorticity ω and the stream function ψ which are related to the velocity field of the fluid. The string function acts as a stream function for the large-scale potential energy flow under the combined beta and topographic effects. 7 stream function-vorticity navier-stokes equation 17 2. When proper boundary conditions are imposed (no vorticity), the solution always exists and is unique. An important concept in the study of aerodynamics concerns the idea of streamlines. 294326e+9, 1. 1 : JAN 9 : Website tour, course requirements, content, grading, schedule, labs, reporting, comp. Indexing of unknowns. The ﬂuid height at the z-axis decreases monotonically as a. In the above MATLAB program, a function, x = gauss_siedel( A ,B ), is initially defined. 2 The stream function and divergence-free flow. Functions Fluid is contained in a square domain with Dirichlet boundary conditions on all sides, with three stationary sides and one moving side (with velocity tangent to the side). Ducted turbulent flows with varying wall shapes are formulated with the Navier-Stokes equations written in terms of vorticity and stream function. 3 Helmholtz’s Theorems 159 10. Numerical results suggest that solving the stream-function-vorticity equations seems more efficient than solving the fourth-order stream function equation. The MATLAB plot is given below: It represents stagnation flow toward a bump. Matlab was used to track the LEV location and calculate the vorticity and circulation magnitudes. The scheme is implemented in the elliptic coordinates with appropriate boundary conditions to account for the wing motion (Wang, 2000a,b). Presents the theoretical backgrounds for the analysis and design of simple feedback control systems, differential equations, and Laplace transforms. Matlab code for fluid flow. 5 and t = 1 s, and for steady state (steady state is reached when, for two successive time instances t n + 1 and t n for both vorticity and stream function field values, N R M S E / d t < 10 − 8). 2 Kinematic Results Concerning Voriticity. 1 Example: square mapping. An icon used to represent a menu that can be toggled by interacting with this icon. The video shows color contour plots of the temperature and streamwise vorticity, in a transverse plane 6 cylinder diameters down stream of the junction, as a function of time during a “typical” periodic cycle. In our case, λ=ρ. , vorticity, Q, lambda2, shearStress, etc. clc clear % X=-1:. Using MATLAB and principles of Computational fluid dynamics (CFD) we had to simulate the flow into and out of the room taking into account the boundary conditions in the walls,door and exit. m (Implements the upwind scheme) 7. Onishi; artificial boundary conditions for vorticity automatically generated. When computers appeared, the indicated problem remained, as before, the focus of attention of specialists developing computational methods. The pressure calculation in the stream vorticity approach uses the stream function to calculate the value of pressure at all the grid points. The stream function (Ψ) and vorticity (Ω) are introduced to fulfill v z = 1 r ∂ Ψ ∂ r, v r = − 1 r ∂ Ψ ∂ z, and Ω = ∂ v r ∂ z − ∂ v z ∂ r. This MATLAB function computes the curl (curlx, curly, curlz) and angular velocity (cav) perpendicular to the flow (in radians per time unit) of a 3-D vector field U, V, W. b the collision frequency function f 1 the eigenfunction of fundamental disturbances f 2 the eigenfunction of sub-harmonic disturbances c the stream function of disturbances r the ﬂuid density r p the particle density n the kinetic viscosity y the initial vorticity thickness of mixing layer u the particle volume u g the geometric mean volume. 8 governing equations in cavity flow 17 2. 17 Find the position (x, y) on the upper surface of the half-body in Fig. I'm analyzing data from a PIV lab and I need to find vorticity and plot it with respect to the y coordinates. Navier–Stokes equation in vorticity-stream function formulation (E and Liu, 1996). The stream function and the vorticity are related by! = @uy @x (FFT). m (Solves the velocity function) 4. 1-5: 1 2 : 14 : Problem statements in. 4 Conformal Mapping. transformed into vorticity–stream function variables for economies of computational resources were obtained. The equation used for the calculation of the pressure is shown below. Physical interpretations for individual terms. The MATLAB plot is given below: It represents stagnation flow toward a bump. Two different solutions can be added linearly and the sum represent also a solution with the corresponding boundary conditions. I'm trying to get matlab to draw a pretty stream function for me. A difficulty with the streamfunction-vorticity formulation for viscous flows is that there are no boundary conditions specified for the vorticity. 4 p 50-56), stream function (Ku 3. s for the stream function is quite simple from its definition in terms of the velocity field. 9 Further Reading 44 References 45 3 Circulation. Particle in Cell Consulting LLC is a small California-based company specializing in the development of high performance computer simulation programs for electric propulsion plasma thrusters, plume-spacecraft interactions, and molecular and particulate contamination transport. Quiver3 Matlab Quiver3 Matlab. Many of the utilities perform vector and tensor operations to compute derived variables, e. In [6, 8, 15] a 2D axially-symmetric ideal, compressible swirling flow with simple chemical raction is descibed by four Euler and two reaction-diffusion equations. Holomorphic functions, complex integral theorems, Runge's theorem, residue theorem, Laurent series, conformal mapping, harmonic functions. stream function distributions when the source term is applied. A streamline is a path traced out by a massless particle as it moves with the flow. * MATLAB 5 (MATLAB R13), * Basic (Visual Basic 6/7 3). Navier stokes matlab. Title: Microsoft Word - outline12. Aerodynamic forces and moments. A circulating flow extending the. Howden and Watts (1999) objectively mapped spatial derivatives of velocity in the Gulf Stream by applying a linear operator to the correlation functions (Bretherton et al. Vorticity equation in different co-ordinates. Theory of Functions of a Complex Variable II. Barragy and Carey used the p version of FEM [1], while Marchi et al. ; L=10; M=10; N=1; LX=1; LY=1; DX=LX/L; DY=LY/M; for i=1:L for j=1:M; for k=1:N. عرض ملف Navheen Shanmugham Murugesan الشخصي على LinkedIn، أكبر شبكة للمحترفين في العالم. You just rewrite the continuity (the divergence-free constraint) and momentum equation (applying the curl). m (Solves the stream vorticity function) 3. The equation used for the calculation of the pressure is shown below. The equations are solved by using matlab Version 6. The vorticity vector at a point is de ned as twice the angular velocity and is w = Ñ V (1) which, for two-dimensional ow in x-y plane, is reduced to w z = w kˆ = ¶v ¶x ¶u ¶y (2) For two-dimensional, incompressible ows, a scalar function may be de ned in such a way that the continuity equation is identically satis ed if the velocity. 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. 12 Hyperbolic equations: Burger's equation. As mentioned earlier, the vorticity transport equation is a parabolic PDE and thus any suitable method for parabolic PDE can be used to solve equation (14). Such a numerical scheme decouples the Stokes solver into two Poisson-like solvers at each time stage in the Runge-Kutta temporal discretization. Note that this is quite different from the built-in MATLAB function curl, which won't help you much. In Matlab, this may be done with the following lines of code: 1. Scalar (0th order tensor), usually we consider scalar elds function of space and time p= p(x;y;z;t) Vector (1st order tensor), de ned by direction and magnitude ( u) i = u i If u = 2 4 u v w 3 5then u 2 = v Matrix (2nd order tensor) (A) ij = A ij If A = 2 4 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 3 5then A 23 = a 23 Kronecker delta (2nd. To a first approximation, this term is equal to fσ t (we replace the vorticity by the planetary vorticity). txt) or view presentation slides online. 8 Fourier Series in MATLAB. ; L=10; M=10; N=1; LX=1; LY=1; DX=LX/L; DY=LY/M; for i=1:L for j=1:M; for k=1:N. Presents the theoretical backgrounds for the analysis and design of simple feedback control systems, differential equations, and Laplace transforms. 3 Defining and Plotting Functions in MATLAB. Finite Volume Methods to Solve Incompressible N-S Equations. nptelhrd 4,776 views. The implementation of the scalar equation is only compulsive for PhD students. only the -component can be non-zero. You just rewrite the continuity (the divergence-free constraint) and momentum equation (applying the curl). Potential flows involving sources and sinks, doublets, and vortices. The results were in a very good agreement. Developed a 2-dimensional incompressible Navier-Stokes solver based on the vorticity-stream function equations, verified against test cases from literature. I've been tasked with modelling this 2D incompressible viscous flow using these boundary conditions and for these values of U₁ and U₂ on a rectangular square grid with 193 by 129 grid points. -Developed a 2D Computational Fluid Dynamics(CFD) Solver program for a backward facing step flow problem by implementing vorticity stream-function approach using FORTRAN -Developed MATLAB programs for correctly plotting the results generated by the previous FORTRAN programs -Performed grid dependence studies on all the above. 1 2 2 + = (+) 2 2 9) Grid used Cavity has been discretized into small elements. The scheme is implemented in the elliptic coordinates with appropriate boundary conditions to account for the wing motion (Wang, 2000a,b). Kammalapati, Propagation of Shock on NREL Phase VI Wind Turbine Airfoil under Compressible Flow , Journal of Renewable Energy, vol. A diﬀerent form of equations can be scary at the beginning but, mathematically, we have only two variables which ha-ve to be obtained during computations: stream vorticity vector ζand stream. The harmonic conjugate psi(x,y) to the velocity potential, in fluid mechanics, is known as the stream function. 2 Vorticity 91 4. Posted by 4 years ago. 5 A quantitative discussion 40 8 THE MUNK MODEL AND PEDLOSKY’S DERIVATION 45 8. The boundary layer theory is shown and flow past immersed bodies are studied: concepts of drag and lift forces are. Numerical Methods for Solving the Vorticity-Stream function Equations ME535 Final Project Frank Bremer June 5, 2008 Abstract A commonly used atmospheric model involves the vorticity and stream functions. m (Solves the stream vorticity function) 3. 1 Blasius’s theorem. 7 Conditional Statements in MATLAB. 13 Vorticity Dynamics. 4 p 50-56), stream function (Ku 3. Ω is simply frame rotation (taken to be known for the purposes of this question). Equation is solved for stream function in the first half‐time step n + 1 2. m flow past half cylinder, time-dependent, demo3. Stream function; the magnetization direction perpendicular to the flow direction Fig. Eulerian descriptions, material derivative, motion of fluid particles, streamlines, stress and strain, vorticity, circulation, stream function; cylindrical and polar coordinates HW 1. A circulating flow extending the. Pseudo-spectral methods are very well conditioned and some operations can be safely performed in single precision without affecting the overall quality of the solution. Using the MATLAB software was locally refined near the immersed boundaries and regions of vorticity, making a contour map of the stream function, since the. 2: In two dimensional incompressible flow the vorticity remains constant in any given fluid element throughout the subsequent evolution. m flow past cylinder, time-dependent,. Diﬃculties in modelling ﬂuid ﬂow common to all of these methods are highlighted with a set of test problems and possible methods of ameliorating these deﬁciencies discussed. Boffetta and G. Torn-2006-BCL doi ucd perturbations in BC needed Fujita-2007-SDA doi ucd using different BC on different ensemble members, no mention what happens in analysis WRF boundary conditions. Flow separation and stream function are also determined. The Stream Function Wave Theory is widely accepted to best cope with regular waves and which in turn has the advantage of leading to a fast computation time, thus it is commonly used to predict kinematics for such kind of waves. 1 The Munk model 45 8. I see no evidence that matlab has made a provision for this in the 'curl' function. where the vorticity is given as, the horizontal velocity, U and vertical velocity, V are given as (4) and x and y are the Cartesian coordinates. The compressible reactive Navier–Stokes equations in terms of stream function, vorticity, circulation are used. 4 Line Vortices 160 10. 7 Further Reading 167. Stream function()의 정의와 stream function으로 알 수 있는 유체의 정보는? 9. Magnetic field, B and Stream function ψ taking them as function of x, y and. syms x y z real F = [ cos(x+2*y), sin(x-2*y) ]; vectorfield(F,-2:. (b) The vorticity ﬁeld of a ﬂowing soap ﬁlm. 1 Basic Assumptions of Lifting Line. Basics of finite-difference methods of discretization, explicit and implicit schemes, grid sizes, stability, and convergence. In this article, we first develop a semi-discretized Crank–Nicolson format about time for the two-dimensional non-stationary Stokes equations about vorticity–stream functions and analyze the existence, uniqueness, stability, and convergence of the semi-discretized Crank–Nicolson solutions. The governing equations above are solved on a grid mesh with 51 x 51 grid points for a square driven cavity using the SOR method and. description } Lid driven cavity matlab code pdf Lid driven cavity matlab code pdf. 2 Vorticity 91 4. Experimental studies. here centered on timestep n. and the stream function is unique by the additional condition Z = 0: The vorticity equation (2) constitute an in nite dimensional Lie{Poisson system (cf. 9 Conclusions. The streak amplitude is deﬁned by, A(x. The stream function is defined as the flux across the line O -P. m (Computes Errors) Terminologies used in code$^2$ 1. Stream Function. 1 Composition of Analytic Functions. 4 Plane Ideal Flows 35 2. The stream function (see Section 2. Description [curlx,curly,curlz,cav] = curl(X,Y,Z,U,V,W) computes the curl (curlx, curly, curlz) and angular velocity (cav) perpendicular to the flow (in radians per time unit) of a 3-D vector field U, V, W. Lines along which is constant are streamlines (details). vorticity의 의미는? 7. Physical interpretations for individual terms. I will be using MATLAB in the classroom, but students are not restricted in their Stream-Function Vorticity PROJECT. In other coordinate systems, the curl requires corresponding modifications. Posted by 4 years ago. Stream-function vorticity form Time-dependent form after H. A uniform grid has been used. Navier–Stokes equation in vorticity-stream function formulation (E and Liu, 1996). Spherefun is the part of Chebfun for computing with functions defined on the surface of the unit sphere. m (Implements the Boundary conditions) 6. This MATLAB function computes the curl (curlx, curly, curlz) and angular velocity (cav) perpendicular to the flow (in radians per time unit) of a 3-D vector field U, V, W. Numerically solves the vorticity stream function formulation of the Navier Stokes equation for a square obstacle in a channel. 11 discretization 22 chapter 3 methodology 3. Pot_vort : Horizontal slice of the vertical component of Ertel’s potential vorticity:. Dubois F, Salaün M, Salmon S. The time behavior is evaluated using a Runge-Kutta ordinary differential equation solver. From above equation, we observe that field is changed and vice versa. warrant the accuracy of the text or exercises in this book. Then what we get is a discretized stream function $\,\psi(x,y)$. Solves the 2D incompressible Navier-Stokes equations in vorticity/stream function formulation on the torus, using spectral methods. das) for this Dataset Attributes { time { String _CoordinateAxisType "Time"; Float64 actual_range 1. Navier stokes matlab. Defining stream function and velocity potential function. Results, details of the numerical algorithms, as well as numerical verification techniques and validation with sources from the literature will be presented. During the austral summer to fall, extreme cyclonic vorticity occurs most frequently at the upper level jet stream core, while during the austral winter to spring, there is a main peak on the poleward flank of the subtropical jet and a secondary peak on the poleward flank of the eddy‐driven jet. In particular, keep track of the computationl speed of each method. ψ is termed the stream function, and is associated (in this case of two-dimensional flow) with the name of Lagrange. [5marks] (ii) A complex potential is given by w= A 2π log(z), Aa positive real constant. 2013, Article ID 653103, 2013. The calculations are made by a computer program, written in MATLAB. In this post I am going to write a (hopefully) simple code in matlab to solve the cavity flow problem using the vorticity stream function formulation. x Plotted vorticity and visualized the flow characteristics contours of stream function Arizona State University Aug - Dec 2014. This Demonstration illustrates the steady eddy structure in a 2D driven cavity problem (see [2] and [3]). 1 Comparison of two boundary condition assignments for the Poisson equation. Vorticity -stream function formulation: • Poisson equation to compute stream function from vorticity 2D periodic domain RK4 time advancement Hyper-viscosity Matlab scripts from Univ. The time behavior is evaluated using a Runge-Kutta ordinary diﬀerential equation solver. MATLAB Files Projects Solves the 2D incompressible Navier-Stokes equations in vorticity/stream function formulation on the torus, using spectral methods. 8 governing equations in cavity flow 17 2. Problem 2: See Examples for 14. Library of Congress Cataloging-in-Publication Data. Learn more about streamfunction, poisson equation, vorticity, velocity field. 5 Segmented Vortex Filaments 161 10. Torn-2006-BCL doi ucd perturbations in BC needed Fujita-2007-SDA doi ucd using different BC on different ensemble members, no mention what happens in analysis WRF boundary conditions. List of Figures 1. This e ectively sets up a no-normal ow through the outer boundary. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled advanced technological achievements, including aircraft, combustion engines, satellites, and electrical power. 4 Vorticity 39 7. simulation was done using MATLAB and the grid size is 161×161 in a staggered grid arrangement. 4 Plane Ideal Flows 35 2. Diﬃculties in modelling ﬂuid ﬂow common to all of these methods are highlighted with a set of test problems and possible methods of ameliorating these deﬁciencies discussed. 1997 ; Meinen and Luther. 7 Conditional Statements in MATLAB. Garewal Text book of Engg. 0 introduction 24 3. Fractal Mysticism 2. 4 Vorticity-Stream Function ap-proach Vorticity-Stream Function approach to two-dimensional problem of solving Navier-Stokes equations is rather easy. m to build the energy measure matrix in wall-normal velocity/wall-normal vorticity formlation. Bernardhas35years’experienceteachinggraduate. For instance, one may plot the stream function, the individual ﬂ ow components, deformation, or vorticity. s for the stream function is quite simple from its definition in terms of the velocity field. The Stream Function Wave Theory is widely accepted to best cope with regular waves and which in turn has the advantage of leading to a fast computation time, thus it is commonly used to predict kinematics for such kind of waves. (b) The vorticity ﬁeld of a ﬂowing soap ﬁlm. Vorticity, circulation, the stream function and the velocity potential. 9 Conclusions. The Dataset Attribute Structure (. 2 Vortex Lines, Vortex Tubes, and Vortex Filaments 159 10. This tutorial was contributed by Francis Poulin, based on code from Colin Cotter. In this post I am going to write a (hopefully) simple code in matlab to solve the cavity flow problem using the vorticity stream function formulation. x ct/,wherekand care the wavenumber and wave speed, respectively, with c D !=k, !being the fre-quency of the disturbance. ψ is termed the stream function, and is associated (in this case of two-dimensional flow) with the name of Lagrange. First, the process of resonant interaction between two progressive interfacial waves is studied. There is a lot going in to this derivation but it is intended to give the stream function and therefore velocity at equilibrium of a rotating system. 334 Snell Engineering Center 617. the problem is, after the coding, i can't get the nice solution. Stream-vorticity implementation will eliminate the pressure term from governing equation by cross-differentiation of the x-momentum and y-momentum equation and makes the problem easy to construct numerical schemes. $\begingroup$ @R. 3 Uniform stream For a uniform ﬂow along the z-axis, u = (0,0,U), the velocity potential φ(z) = Uz. i'm sorry to bother you all, but can you send me the code that's working because my code isn't. You just rewrite the continuity (the divergence-free constraint) and momentum equation (applying the curl). equation for the potential and/or the stream function is a linear boundary value problem. 5a for which the local velocity equals the uniform stream velocity. Path lines, stream functions, and vorticity are treated and calculated, too. 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. $\begingroup$ @R. Pseudo-spectral methods are very well conditioned and some operations can be safely performed in single precision without affecting the overall quality of the solution. Calculate the velocity eld: u(x;y) and v(x;y). X, Y, and Z must have the same number of elements, as if produced by meshgrid. The stream function i design it to be dimensionless and the same for the vorticity. Solving this system for each time twe plot the. other GUI is a MATLAB box, that requires input from the user for the speed of the uniform flow over the airfoil, the airfoil angle of attack, and the vorticity of the flow field, due to the object rotating in the flow field. independent vorticity ﬁeld (45) (i. Treats system modeling, linear approximations, transfer functions, and block diagrams; and transient and frequency response and stability-frequency domain and root locus methods. I have position data for x and y, as well as velocity data in the x and y directions. using on the RHS the value of η at time n. Use coordinates, rather than : , still. matlab curve-fitting procedures, according to the given point, you can achieve surface fitting,% This script file is designed to beused in cell mode% from the matlab Editor, or best ofall, use the publish% to HTML feature from the matlabeditor. Magnetic field, B and Stream function ψ taking them as function of x, y and. Essentially, the system is composed of the vorticity transport equation (9) and the Poisson equation for streamfunction (15). The equations are solved by using matlab Version 6. The simplest way to avoid the case is to eliminate the vorticity and turn to a single nonlinear biharmonic equation for the stream function. There is a nice introduction to the functionality here. only the -component can be non-zero. Finite Element Methods to Solve Incompressible N-S Equations. The results were in a very good agreement. This equation can be solved using the stream function-vorticity method for 2D incompressible flow , which can also be applied to 3D flow with axial symmetry. Explanation of curvature and shear vorticity with specific examples. 15) Example 6. Besides, stream function and vorticity values in addition to the location of primary and secondary vortices’ centers inside the square cavity have been revealed at low and intermediate Reynolds. Howden and Watts (1999) objectively mapped spatial derivatives of velocity in the Gulf Stream by applying a linear operator to the correlation functions (Bretherton et al. The MATLAB scripts solve the Euler equation in vorticity-stream function using a pseudo-spectral method. I have been asked. The two velocity-based approaches differ in how they calculate the stream function from velocity data: The ﬁrst approach uses modal decomposition, while the second obtains the stream function using path integrals. 6: Vorticity. [5marks] (ii) A complex potential is given by w= A 2π log(z), Aa positive real constant. other GUI is a MATLAB box, that requires input from the user for the speed of the uniform flow over the airfoil, the airfoil angle of attack, and the vorticity of the flow field, due to the object rotating in the flow field. - Special thanks to my previous teachers and instructors who introduced me to the world of fluid mechanics. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. distributed vorticity drives the velocity that advects that same vorticity distribution. Question Cagtegory: Fluid dynamics. 9 Further Reading 44 References 45 3 Circulation. This requires the solution of a relatively small number of perturbed eigenvalue prob-lems determined from a special set of parameter values, using sparse-grid methods [1, 22]. Using MATLAB and principles of Computational fluid dynamics (CFD) we had to simulate the flow into and out of the room taking into account the boundary conditions in the walls,door and exit. approaches obtain the energy ﬂux in terms of the stream function [7,19], obviating the need for the pressure ﬁeld. Numerical solutions present the average Nusselt number variation as well as streamlines, isotherms and. using on the RHS the value of η at time n. m flow past cylinder, time-dependent,. The stream function (see Section 2. 4 and the disappearance of vorticity in Fig. After computing initial values for the vorticity field, the iteration starts with solving for the streamfunction using the Jacobi Iteraition. Week 3: Solving momentum and continuity equations in case of constant viscosity with stream function/vorticity formulation. Based on the geometry of the airfoil, the code then builds a far-field rectangular. Subseuqently, the vorticity on the boundary is found and the transport equation solved by an upwind scheme to counter numerical diffusion. I thus calculate the vorticity $\omega$ (such that $\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$) and then solve the Poisson equation $abla^2 \psi=\omega$. Numerical Methods for Solving the Vorticity-Stream function Equations ME535 Final Project Frank Bremer June 5, 2008 Abstract A commonly used atmospheric model involves the vorticity and stream functions. Developed a 2-dimensional incompressible Navier-Stokes solver based on the vorticity-stream function equations, verified against test cases from literature. The stream function and vorticity equations can be solved using the finite difference method. stream function distributions when the source term is applied. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java™. Matlab code for Solving Navier Stokes equations in Vorticity- Stream function formulation May 2015 – Dec 2015. of Washington (Amath 571) for evolution of isolated elliptic vortex and 2D isotropic turbulence. 12 Hyperbolic equations: Burger's equation. Fractal Metaphor 3. Note that for compressible flow this is not the case in regions of large entropy gradient. Numerical Solution of vorticity, stream function and pressure are also obtained. NavierStokes. whole height was observed for shallow and square cavities. 4 Solution of Navier-Stokes Equations 598 19. Learn more about streamfunction, poisson equation, vorticity, velocity field. Streamfunction from vorticity or velocity field. 723 - Computational methods. MAC-SIMPLE Methods to solve Incompressible Flows and Heat Transfer. Requires the relative vorticity components and the potential density, and produces one output file with the potential vorticity (file prefix is PV for the full term and splPV for the planetary component). pdf), Text File (. 5 Vorticity in Barotropic Fluids 106 4. Three different Fluid Solvers have been implemented to handle and compare results: SIMPLE (Semi Implicit Method for Pressure Linked Equations), SIMPLER (Semi Implicit Method for Pressure Linked Equations Revised) and Vorticity - Stream Function Approach. Scaling hypothesis leading to extended self-similarity in turbulence. Here Qis the z-component of the vorticity of the circular ﬂow. gate function defined by interpolation. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java™. The stream function and vorticity equations can be solved using the finite difference method. [15] and Benjamin et al. The remotely operated vehicles (ROVs) are important to provide the technology support for both the traditional offshore structures and rapidly-growing renewable energy facilities during their full-lifecycles, such as site survey, installation, inspection, maintenance and repair. they measured the free surface height at the z-axis as a function of the Reynolds number Re (the Reynolds number is deﬁned in Sec. Stream function()의 정의와 stream function으로 알 수 있는 유체의 정보는? 9. Filter by. 1 The Circulation Theorem 86 4. 1 Answer to A two-dimensional incompressible flow is given by the velocity field V= 3yi+ 2xj, in arbitrary units. Then what we get is a discretized stream function $\,\psi(x,y)$. Potential Vorticity Conservation From the Shallow Water Equations, we derive the principle of conservation of potential vorticity: d dt + f h = 0: where is therelative vorticity, f is theplanetary Vorticity and h is theﬂuid depth. ENGINEERING MATHEMATICS (M-101) Higher engineering mathematics by B. 4 Line Vortices 160 10. Vorticity, induced velocity and irrotational flow. Aerodynamics of inviscid incompressible flows. i'm sorry to bother you all, but can you send me the code that's working because my code isn't. Cross differentiating (11. 4 Vorticity-Stream Function ap-proach Vorticity-Stream Function approach to two-dimensional problem of solving Navier-Stokes equations is rather easy. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations. By making some calculations we can find out. Other readers will always be interested in your opinion of the books you've read. w = $\omega$(Vorticity). independent vorticity ﬁeld (45) (i. These can be visualised as contour plots, colour-coded“surf” plots, or the ﬁ elds can be plotted along the polar axis or along a line in the equatorial plane. Finite difference method is used to discretize the governing partial differential equations. Concept of planetary vorticity. Equation is solved for stream function at the second half‐time step (n + 1). Howden and Watts (1999) objectively mapped spatial derivatives of velocity in the Gulf Stream by applying a linear operator to the correlation functions (Bretherton et al. pdf: reference module 5: 13: Solution of Navier-Stokes Equations in Curvilinear Coordinates: reference_mod6. True, but for matlab's 'curl' function only if cartesian coordinates are being used. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations. This book is served as a reference text to meet the needs of advanced scientists and research engineers who seek for their own computational fluid dynamics (CFD) skills to solve a variety of fluid flow problems. The remotely operated vehicles (ROVs) are important to provide the technology support for both the traditional offshore structures and rapidly-growing renewable energy facilities during their full-lifecycles, such as site survey, installation, inspection, maintenance and repair. visualized using Matlab code on the basis of a suite of equations. Mathematical description of pertinent physical phenomena. Solving this system for each time twe plot the. Concepts of stream function, vorticity and velocity potential are also introduced. 8 Fourier Series in MATLAB. conservation of mass and momentum. 2 Stream Function Formulation (Using Variational Approach) 592 19. Using MATLAB and principles of Computational fluid dynamics (CFD) we had to simulate the flow into and out of the room taking into account the boundary conditions in the walls,door and exit. 7 Velocity Potential and the Unsteady Bernoulli Equation. The stream function equation is discretized using the standard central difference, and can be solved using an iterative elliptic solver, such as Jacobi or Gauss-Seidel. The usefulness of the stream function lies in the fact that the flow velocity components in the x- and y- direct. This can be overcome using a linear correction technique known as the influence matrix method. (1) in terms ofy, and each term in Eq. only the -component can be non-zero. Numerical Methods for Solving the Vorticity-Stream function Equations ME535 Final Project Frank Bremer June 5, 2008 Abstract A commonly used atmospheric model involves the vorticity and stream functions. Lines along which is constant are streamlines (details). 2 Step function constructed by applying the. Find books. Quasi-Geostrophic Model¶. The calculations were performed by brigades of specialists without the use of computers. Hey all, This is my first time posting here so wondering how helpful this sub is. 1 Introduction 591 19. hk das engineering maths PDF download. 2 The stream function 36 7. - Special thanks to my previous teachers and instructors who introduced me to the world of fluid mechanics. The arrays X, Y, and Z, which define the coordinates for U, V, and W, must be monotonic, but do not need to be uniformly spaced. Heat conduction. In other coordinate systems, the curl requires corresponding modifications. The usefulness of the stream function lies in the fact that the flow velocity components in the x- and y- direct. Arrays are initialized (set to zero) and boundary conditions for stream function set up. Incompressible plane flows, Stream function and vorticity equations, Conservative form and normalizing systems, Method for solving vorticity transport equation, Basic finite difference forms, Conservative property, Convergence and stability analysis, Explicit and implicit methods, Stream function equation and boundary conditions, Schemes for. 2 Pedlosky’s derivation 46 v. niques rely on the vorticity-stream function formulas, see for example Ghia et al. Matlab was used to track the LEV location and calculate the vorticity and circulation magnitudes. Equation is solved for stream function at the second half‐time step (n + 1). in matlab: Plots a histogram with a gaussian fit to the data. 1 Answer to A two-dimensional incompressible flow is given by the velocity field V= 3yi+ 2xj, in arbitrary units. vorticity are derived in the previous Section. Methods Appl. Numerical solutions present the average Nusselt number variation as well as streamlines, isotherms and. Hey all, This is my first time posting here so wondering how helpful this sub is. asymptotically horizontal at large distances. Having demonstrated that the stream-function-vorticity approach could be used to simulate low-speed two-dimensional flow problems, Prof. This Demonstration illustrates the steady eddy structure in a 2D driven cavity problem (see [2] and [3]). Lid driven cavity matlab code pdf { twitter. The stream function is defined as the flux across the line O -P. The effects of dimensionless Hartmann number, Rayleigh number, the number of undulations, amplitude of wave, and the location of magnetic source are visualized in streamlines and isotherms as. 71) The appropriate boundary condition at the surface of the cylinder is simply that the normal fluid velocity there be zero, because the fluid must stay in contact with the cylinder, but cannot penetrate its surface. Numerically solves the vorticity stream function formulation of the Navier Stokes equation for a square obstacle in a channel. Numerical Methods for Solving the Vorticity-Stream function Equations ME535 Final Project Frank Bremer June 5, 2008 Abstract A commonly used atmospheric model involves the vorticity and stream functions. startx, starty, startz define the starting positions of the streamlines. Having demonstrated that the stream-function-vorticity approach could be used to simulate low-speed two-dimensional flow problems, Prof. Weeks 4: Staggered grid for formulating momentum and continuity equations. Arrays are initialized (set to zero) and boundary conditions for stream function set up. The string function acts as a stream function for the large-scale potential energy flow under the combined beta and topographic effects. ; L=10; M=10; N=1; LX=1; LY=1; DX=LX/L; DY=LY/M; for i=1:L for j=1:M; for k=1:N. 4 Line Vortices 160 10. In previous experiments where objective mapping was used to compute streamfunction and velocity from an array of CPIES (e. This function accepts GPU arrays, but does not run on a GPU. Science 290 (5500), 2319-2323 You may access the full text of the document here, or visit the ISOMAP Homepage for further details, Matlab code and data sets. I never knew there was a metaphysical aspect to Fractals. The fluid motion is studied in stream function-vorticity formulation. Use Matlab to find numerical solutions of certain more complicated flow situations Understand the basic features of subsonic airfoils, including lift, drag and stall Apply shock expansion theory to the solution of flow in a variety of situations including prediction of lift and drag of two-dimensional bodies in supersonic flow. It is also a physical analog for 2-D flow of incompressible fluids through porous media by. matlab potential flow solver and simulator. Scale analysis of vorticity equation. Concepts of stream function, vorticity and velocity potential are also introduced. 4 and the disappearance of vorticity in Fig. The 2D stream function-vorticity formulation is a standard section in any textbook of CFD and is a good exercise for a student. Scaling hypothesis leading to extended self-similarity in turbulence. Numerically solves the vorticity stream function formulation of the Navier Stokes equation for a square obstacle in a channel. Vorticity 2d - Cfd - Matlab. The stream function (see Section 2. ψ is termed the stream function, and is associated (in this case of two-dimensional flow) with the name of Lagrange. Numerical results suggest that solving the stream-function-vorticity equations seems more efficient than solving the fourth-order stream function equation. $\begingroup$ @R. The Navier-Stokes equations in vorticity-stream function form are:! Elliptic equation! Advection/diffusion equation! The vorticity/streamfunction equations:!. vorticity are derived in the previous Section. A circulating flow extending the. The equation used for the calculation of the pressure is shown below. Magnetic field, B and Stream function ψ taking them as function of x, y and. Source, sink,doublet, vortex, spiral flow, flow past a cylinder with circulation-Magnus effect-Joukowski theorem. 11 Lifting Line Theory 169 11. We introduce a new recasting of the planar Navier-Stokes equations as a ﬁrst-order system based on stress and stream. Other readers will always be interested in your opinion of the books you've read. This includes not only image processing functions (ranging from contour tracing to Fast Fourier Transforms), and data analysis functions (such as statistics, least squares fits), to numerical solution of the equations of motion (e. Sorry I can’t help more. Quiver3 Matlab Quiver3 Matlab. Fluid statics/buoyancy force. 5a for which the local velocity equals the uniform stream velocity. vorticity-stream function formulation in dimensionless form as. The vorticity is related to the velocity by introducing a 2D stream-function: r2 (x;y) = !(x;y) With the boundary conditions: = 0 On the outside of the domain. 1 Review of Complex Variables 35 2. Learn more about streamfunction, poisson equation, vorticity, velocity field. This tutorial was contributed by Francis Poulin, based on code from Colin Cotter. ω - Vorticity Ω - Dimensionless vorticity ψ - Stream function Ψ - Dimensionless stream function ∇ - Nabla operator Superscript n - Current value n + 1 - Next step value * - Non advection phase value Subscript i - x direction node j - y direction node max i - x direction maximum node. True, but for matlab's 'curl' function only if cartesian coordinates are being used. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. Stream functions: A stream function, Ψ, is the function used to plot streamlines and tell us the speed of the flow/the volume flow rate. MATLAB Files Projects Solves the 2D incompressible Navier-Stokes equations in vorticity/stream function formulation on the torus, using spectral methods. Such a solution satisfies the discrete form of the continuity equation exactly and coincides with that obtained by solving the vorticity-stream function equations on the same grid,14 so that it can be considered as the best possible second-order-accurate finite difference solution. streamlines are tangent to the flow velocity vector of the flow, the value of the stream function must be constant along a streamline. Remember to specify all the units of your results. Here are three ways to illustrate a vector field beside plotting arrows. Matlab code for Solving Navier Stokes equations in Vorticity- Stream function formulation May 2015 – Dec 2015. Note that the Laplacian function in cylindrical coordinates is often written in the more “compact” manner on the left side of Eq. I have used matplotlib's plot_streamlines function in the past to create streamline plots. velocityField postProcessing tools Post-processing velocity fields. We consider an isopycnal layer with the interval [ σ 1 , σ 2 ]. Finally, U and V velocities are extracted from the streamfunction. Many of the utilities perform vector and tensor operations to compute derived variables, e. mathematical formulations has been used, including stream function and vorticity ( ψ-ω); stream function and velocity ( ψ-V); lattice Boltzmann equation (LBE), and the Navier-Stokes equations ( u-v-p). 1-5: 1 2 : 14 : Problem statements in. The software listed above does not constitute a fulllistofavailableoptions,asotherprogramssuchas Metrowerks Codewarrior6 and Sun Java Studio do exist for compilation of the different languages. Time-independent form as elliptic system after Barragy-Carey. Library of Congress Cataloging-in-Publication Data. 1 2 2 + = (+) 2 2 9) Grid used Cavity has been discretized into small elements. The pressure tenn was eliminated by differentiating each term in Eq. Find books. only the -component can be non-zero. nptelhrd 4,776 views. b the collision frequency function f 1 the eigenfunction of fundamental disturbances f 2 the eigenfunction of sub-harmonic disturbances c the stream function of disturbances r the ﬂuid density r p the particle density n the kinetic viscosity y the initial vorticity thickness of mixing layer u the particle volume u g the geometric mean volume. employing the stream and vortices functions [7]. If you want more control over the creation of the plot this thread at StackOverflow offers more flexible techniques (and source code) for creating streamline plots in Python. m (Computes Errors) Terminologies used in code$^2$ 1. Grids to solve the stream function can typically define a compact computational domain with. clc clear N=100; Lx=1; Ly=1; dx=Lx/N; dy=Ly/N; Area=dx*dy; for i=1:N; du(i)=0. We consider an isopycnal layer with the interval [ σ 1 , σ 2 ]. This project aims at simulating lid driven cavity flow problem using package MATLAB. Developed MATLAB code for analysis of 2D driven lid flow using Vorticity/Stream Function method which fall under FDM. Solve the lid driven cavity flow using vorticity-stream function formulation. Fractal Mysticism 2. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. Solution of algebraic equations, relaxation. 1-5: 1 2 : 14 : Problem statements in. 4 s along lines passing through the vortex core center while parallel to r and z axis. Stream Function. Figure 4(a) shows a. Solves the 2D incompressible Navier-Stokes equations in vorticity/stream function formulation on the torus, using spectral methods. I am working with the Vorticity-Stream function formulation to simulate a 2D flow in a channel which is bounded in one direction with no-slip boundary conditions and the other is assumed to be 2pi. This course presents the dynamic governing equations of fluid flow in differential forms: continuity, Navier-Stokes and energy. New Member. function mit18336_spectral_ns2d %%%%% % Navier-Stokes equations in vorticity/stream function formulation on the torus % % Version 1. Mathematical tools such as the potential function, stream function, and complex potential are concisely outlined in Appendix A. The "stream function" is often given by writing the vector potential in a particular form; if ${\bf A} = (0,0,\psi)$, then the resulting flow will be in 2D. As a part of Computational Fluid Dynamics (CFD)course work; MATLAB was used to code. The harmonic conjugate psi(x,y) to the velocity potential, in fluid mechanics, is known as the stream function. with streamlines. Developed MATLAB code for analysis of 2D driven lid flow using Vorticity/Stream Function method which fall under FDM. In particular, keep track of the computationl speed of each method. The geometry of the Rankine Vortex is illustrated in Figure and is defined to be Figure. of Washington (Amath 571) for evolution of isolated elliptic vortex and 2D isotropic turbulence. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Vorticity for solid body rotation. The equations are first discretized in time and space. 10 Vorticity and Vortices 157 10. In this post I am going to write a (hopefully) simple code in matlab to solve the cavity flow problem using the vorticity stream function formulation. mathematical formulations has been used, including stream function and vorticity ( ψ-ω); stream function and velocity ( ψ-V); lattice Boltzmann equation (LBE), and the Navier-Stokes equations ( u-v-p).