The equations of fluid mechanics – the Navier-Stokes equations, are solvable analytically for only a limited number of flows under certain assumptions. The known solutions are extremely useful in helping to understand fluid flow but rarely can they be used directly in engineering analysis or design.

Overview: The equations of fluid mechanics – the Navier-Stokes equations, are solvable analytically for only a limited number of flows under certain assumptions. The known solutions are extremely useful in helping to understand fluid flow but rarely can they be used directly in engineering analysis or design. Therefore it is obligatory to use other approaches in order to solve practical problems. Today, due to the increased computational power of advanced computers, interest in numerical techniques increased dramatically. The solution of the equations of fluid mechanics on computers has become so important that it now occupies the attention of perhaps a third of all researchers in fluid mechanics and the proportion is still increasing. This field is known as computational fluid dynamics (CFD). Contained within it are many subspecialties. We shall discuss only a small subset of methods for solving the equations describing fluid flow and related phenomena. What is CFD? Flows and related phenomena can be described by partial differential equations, which cannot be solved analytically except in special cases. To obtain an approximate solution numerically, we have to use a discretization method which approximates the differential equations by a system of algebraic equations, which can then be solved on a computer. The approximations are applied to small domains in space and/or time so the numerical solution provides results at discrete locations in space and time. Much as the accuracy of experimental data depends on the quality of the tools used, the accuracy of numerical solutions is dependent on the quality of the discretizations used. Contained within the broad field of computational fluid dynamics are activities that cover the range from the automation of well-established engineering design methods to the use of detailed solutions of the Navier-Stokes equations as substitutes for experimental research into the nature of complex flows. Discretization Approaches: 1 Finite Difference Method:This is the oldest method for the numerical solution of PDE’s, believed to have been introduced by Euler in the 18th century. It is also the easiest method to use for simple geometries. The starting point is the conservation equation in differential form. The solution domain is covered by a grid. At each grid point, the differential equation is approximated by replacing the partial derivatives by approximations in terms of the nodal values of the functions. The result is one algebraic equation per grid node, in which the variable value at that and a certain number of neighbor nodes appear as unknowns. Basic Steps in Finite Difference Method: In order to solve a given PDE by numerical methods, the partial differentials of the dependent variable in PDEs must be approximated by finite difference relations (algebraic equations). The solution of a steady PDE in a two-dimensional rectangular domain with initial and boundary conditions. Division of domain in uniform/non-uniform mesh. Formulation of algebraic equations for each grid point/control volume (matrix system Ax = b). Methods for finite difference approximations. Taylor series expansions. Finite Difference by Polynomials. 2 Finite Volume Method: (FV Method)The FV method uses the integral form of the conservation equations as its starting point. The solution domain is subdivided into a finite number of contiguous control volumes (CVs), and the conservation equations are applied to each CV. At the centroid of each CV lies a computational node at which the variable values are to be calculated. Interpolation is used to express variable values at the CV surface in terms of the nodal (CV-center) values. Surface and volume integrals are approximated using suitable quadrature formulae. As a result, one obtains an algebraic equation for each CV, in which a number of neighbor nodal values appear. The FV approach is perhaps the simplest to understand and to program. All terms that need be approximated have the physical meaning which is why it is popular with engineers. The disadvantage of FV methods compared to FD schemes is that methods of order higher than second are more difficult to develop in 3D. This is due to the fact that the FV approach requires three levels of approximation: interpolation, differentiation, and integration. Basic Steps in Finite Volume Method: Divide the continuous domain into a number of discrete subdomains (control volumes) by a grid. The grid defines the boundaries of a control volume, while the computational node lies at the center of each control volume. For each sub-domain, derive governing algebraic equations from the governing differential equations. Obtain a system of algebraic equations from above. Solve the above system of algebraic equations to obtain values of the dependent variables at identified discrete points (computational nodes). The model usually used to describe the free surface flows is based on the two-dimensional Saint-Venant equations. These quite known models in the literature equations are obtained by integrating the vertical dimensional incompressible Navier-Stokes equations under the assumptions of hydrostatic pressure and averaged velocities in the vertical level. Terms from the turbulence, viscosity, and the Coriolis forces are not considered in this study. The system can be set as conservative form: where u and v are the depth-averaged water velocities in x and y direction, h the water depth, g the gravitational acceleration, Sox and Soy are respectively slopes following the direction x and y. They are defined by Sox=dZ/dx and Soy=dZ/dy. Discretization of diffusion fluxes We consider here a diffusive flux F(x,t) of form F(x,t)=−Λ(x,t)∇u(x,t), where Λ(x,t) is a linear operator from Rd to Rd, and u is one of the continuous unknowns (uj)j=1,…, N of the problem. Such an expression of the flux is encountered when using e.g. Fourier's law (heat conduction), Darcy's law (flow in porous media), or Fick's law (chemical diffusion). In some cases, a simple, conservative, and consistent discrete expression for F(n)K,σ may be obtained. This is the case when Λ(x,t) is the identity operator and when the grid satisfies the following "orthogonal property": in each control volume KM , there exists a particular point xK, called the "center" (or centroid) of the control volume, such that, for any adjacent control volume L to K with common face σ, the straight line (xK, xL) is orthogonal to σ. Figure 2: Two control volumes with a common face, satisfying an orthogonality property This orthogonality property is satisfied by several types of meshes, such as triangles, rectangles, Voronoï boxes. Then a simple finite difference expression for the approximation of Approximation of advection terms: Consider the partial differential equation ∂t u(x,t)+∇⋅F(u(x,t),x,t)=0. This is the conservation equation (1) with A(x,t)=u(x,t) , F(x,t)=F(u(x,t),x,t) and S(x,t)=0 . The linear transport equation is a particular case, with F(u,x,t)=uV(x,t) , and V is defined from Ω×[0,T] to Rd . The discrete unknowns are the values u(n)K , which are expected to be approximations of u in the control volumes K∈M at time t(n) . Since the convection flux is of order 0 (no derivative involved) its approximation does not require any assumption on the mesh, contrary to the diffusion flux. Nevertheless, the fluxes must be carefully approximated in order to ensure the stability (and convergence) of the scheme. Consider the flux ϕ(n)K,σ through an interface σ at time t(n) , seen as a function of the solution u Source term linearization: If the source term under consideration is a non-linear function of a conserved variable, the linearization of it is a fundamental approach to discretizing it in the finite volume method. One of the basic rules (Rule 3) required that when the source term is linearized as\begin{equation}S = S_C + S_P \phi_P, \tag{7.6} \label{eq:sourceTerm}\end{equation} the quantity \(S_P\) must not be positive. Now, we return to the topic of source-term linearization to emphasize that often source terms are the cause of divergence of iterations and that proper linearization of the source term frequently holds the key to the attainment of a converged solution. Rule 3: Negative-slope linearization of the source term If we consider the coefficient definitions, it appears that, even if the neighbor coefficients are positive, the center-point coefficient \(a_P\) can become negative via the \(S_P\) term. Of course, the danger can be completely avoided by requiring that \(S_P\) will not be positive. Thus, we formulate Rule 3 as follows: When the source term is linearized as \(\bar{S} = S_C + S_P T_P\), the coefficient \(S_P\) must always be less than or equal to zero. Cited Source References [1] Hess, J.L.; A.M.O. Smith (1967)."Calculation of Potential Flow About Arbitrary Bodies". Retrieved from doi:10.1016/0376-0421(67)90003-6 [2] Carmichael, R.; Erickson, L. (1981)."PAN AIR - A higher order panel method for predicting subsonic or supersonic linear potential flows about arbitrary configurations". Retrieved from doi:10.2514/6.1981-1255 [3] Hess, J.; Friedman, D. (1983)."Analysis of complex inlet configurations using a higher-order panel method". Retrieved from doi:10.2514/6.1983-1828 [4] Raj, Pradeep; Brennan, James E. (1989)."Improvements to an Euler aerodynamic method for transonic flow analysis".Retreived from doi:10.2514/3.45717 [5] Karman, l. (1995)."SPLITFLOW - A 3D unstructured Cartesian/prismatic grid CFD code for complex geometries".Retrieved from doi:10.2514/6.1995-343 [6] Giles, M.; Drela, M.; Thompkins, Jr, W. (1985)."Newton solution of direct and inverse transonic Euler equations".Retrieved from doi:10.2514/6.1985-1530 Credits Jade Teekhasaenee CFD Engineering Manager - Cradle Consulting ThailandThe author had experienced in wide researches of Combustion engine, and Chemical Reaction related. He graduates with a Master degree from Drexel University (Graduate School of Engineering) Philadephia, USA, specialized in Advanced Engineering Simulation.