Introduction to Computational Fluid Dynamics

• This is a quick introduction to the basic concepts underlying CFD, using simple 1D model problems to illustrate the concepts. These concepts extend to more general situations and are worth understanding.

Need for CFD

• Applying fundamental laws of mechanics to a fluid yields governing equations.
• The conservation of mass equation is ∂ρ/∂t + ∇ · (ρv) = 0.
• The conservation of momentum equation is ρ(∂v/∂t + v · ∇v) = -∇p + ρg + ∇ · τij.
• These equations, combined with energy conservation, form a set of coupled, non-linear partial differential equations. Analytical solutions are not possible for most engineering problems.
• CFD provides approximate computer-based solutions to the governing equations for various engineering problems.

Applications of CFD

• CFD is used in industry for various applications.
• It can simulate flow over a vehicle: study interaction of propellers/rotors with aircraft fuselage.
• An example is predicting the pressure field induced by a helicopter rotor interacting with the fuselage in forward flight.
• It can analyze temperature distribution in a mixing manifold, like in the Boeing 767 cabin ventilation system. It helps assess simpler designs without field testing.
• Biomedical engineering uses CFD to study circulatory and respiratory systems. It provides pressure contours and cutaway views of velocity vectors in blood pumps.
• CFD is more cost-effective than physical testing, but complex flow simulations require expertise and error-checking to ensure validated solutions.

Strategy of CFD

• CFD replaces the continuous problem domain with a discrete domain using a grid.
• In the continuous domain, each flow variable is defined at every point.
• Pressure p in a continuous 1D domain is given as p = p(x), 0 < x < 1.
• In the discrete domain, each flow variable is defined only at grid points.
• Pressure in a discrete 1D domain is defined only at N grid points.
• In a CFD solution, flow variables are solved for directly at the grid points and other values can be determined through interpolation.
• Governing partial differential equations and boundary conditions are defined in terms of continuous variables (p, v etc.). They can be approximated in the discrete domain.
• This results in a large set of coupled, algebraic equations in discrete variables solved using repetitive calculations by a digital computer.

Discretization Using The Finite-Difference Method

• A simplified 1D equation(du/dx + um = 0; 0 ≤ x ≤ 1; u(0) = 1) illustrates the fundamental ideas underlying CFD.
• Consider the case where m = 1 (linear equation), and later m = 2 (nonlinear equation).
• A discrete representation of the equation with m = 1 on a grid with four equally-spaced points is given.
• Since the governing equation (du/dx + u_i = 0) is valid at any grid point,the expression of (du/dx); is du/dx = (u_i - u_(i-1))/(Δx) + O(Δx).
• The error in (du/dx); due to neglected terms is called the truncation error. A discrete representation with truncation error O(∆x) is termed first-order accurate.
• From these equations, the discrete equation is derived and leads to the transition from a differential equation to an algebraic equation.
• Deriving discrete equations using Taylor's series expansions is the finite-difference method.
• Commercial CFD codes often use the finite-volume or finite-element methods, better suited for complex geometries.

Discretization Using The Finite-Volume Method

• Airfoil grids often consist of quadrilaterals.
• In the finite-volume method, quadrilaterals are referred to as "cells," and grid points as "nodes”.
• In 2D, triangular cells are possible, and in 3D, cells are hexahedrals, tetrahedrals, or prisms.
• The integral form of conservation equations are applied to the control volume defined by a cell to get the cell's discrete equations.
• The integral form of the continuity equation for steady, incompressible flow is ∫V • n dS = 0, where integration is over the surface S of the control volume.
• This means the net volume flow into the control volume is zero.
• Applying the mass conservation equation gives the discrete form of the continuity equation for the cell: -u1∆y - v2∆x + u3∆y + v4∆x = 0.
• This ensures that net mass flow into the cell is zero, i.e., mass is conserved.
• Unlike the finite-difference method, cell center values are usually solved for directly by inverting the discrete system, but face values are obtained by interpolating the cell-center values at adjacent cells.
• Discrete equations for conservation of momentum and energy are used in similar fashion.
• When using FLUENT, the code finds a solution such that mass, momentum, energy, and other relevant quantities are being conserved for each cell. Code directly solves for flow variable values at cell centers, whereas values at other locations are obtained by suitable interpolation.

Assembly of Discrete System and Application of Boundary Conditions

• The discrete equation is -ui-1 + (1 + ∆x)ui = 0.
• Applying this equation to the 1D grid gives: -u1 + (1 + ∆x) u2 = 0 (i = 2), -u2 + (1 + ∆x) u3 = 0 (i = 3), -u3 + (1 + ∆x) u4 = 0 (i = 4).
• The discrete equation cannot be applied at the left boundary, boundary conditions are applied to get: u1 = 1.
• These equations form a system of four simultaneous algebraic equations represented conveniently in matrix form.

Solution of Discrete System

• In a general situation, discrete equations are applied to grid points or cells within the domain.
• At or near the boundary, a combination of the discrete equations and boundary conditions are applied, giving a system of simultaneous algebraic equations.
• Like other commercial CFD codes, FLUENT has various boundary condition options: velocity inlet, pressure inlet, pressure outlet, etc.
• Proper boundary conditions are important otherwise it might not be doing what is expected which can lead to a wrong result.
• The discrete system can be inverted to obtain the unknowns at the grid points such as when solving for u1, u2, u3 and u4 in turn, which yields specific solutions (e.g., u2 = 3/4).
• The discrete solution on a four-point grid can be compared with an exact solution.
• Error is largest at the right boundary, and in practical CFD application, optimizing matrix inversion is important to minimize CPU time and memory required.
• The inverted matrix is sparse because equations at a grid point or cell contain only quantities at neighboring points or cells. Only non-zero values are stored to minimize memory. An iterative procedure generally inverts the matrix. The iteration determines how close one gets to the true matrix inversion solution.

Grid Convergence

• Expect the error in numerical solution decreases as the number of grid points increases and Ax is reduced and there is better agreement between numerical + exact solutions
• The effect of increasing the number of grid points N on the numerical solution of the 1D problem. The assembly and solution steps were repeated on grids and the resulting discrete system was solved using MATLAB.
• As expected, numerical error declines as grid points increase
• Numerical solutions on different grids that agree within a user-specified tolerance are “grid converged.”
• Grid convergence applies to the finite-volume also where the numerical solution becomes independent of the grid as cell size is reduced the smaller it is.
• It is important to investigate the effect of grid resolution on the CFD solution, but you should never trust a CFD solution unless convinced that grid is converged to an acceptable tolerance level (dependent on the problem).

Dealing with Nonlinearity

• Momentum conservation equation is nonlinear due to the convection term (vVv).. Turbulence and chemical reaction introduce additional nonlinearities.
• Due to highly nonlinear nature for governing equations, difficult to obtain accurate numerical solutions for flows
• Demonstrates nonlinearity by setting m = 2 in a simple 1D example: du/dx + u² = 0; 0 ≤ x ≤ 1; u(0) = 1.
• Corresponding first-order finite-difference approximation, analogous to linear equation du/dx+ u- u_(i-1)/Delta x+ u_=0.

Direct and Iterative Solvers

• Iterations are needed to deal with non-linear terms in governing equations. It is necessary to carry out iterations in practical CFD problems.
• Discrete equation system resulting from finite equation is on our four point grid, the inverted matrix must reduce memory required
• (First solve with u_(gi=i) for the guess value; If neighbouring value at the current iteration level isn't available, the guess is used instead.

Iterative Covergence

• The linearization and matrix inversion errors tend to zero when u_g → u. Iterations are continued until some selected measure difference which is referred to as the residual, is small enough. The residual R is the RMS value of the difference between u and u_g on the grid can define as such. This leads to iterative convergence. The variations of the residual in this MATLAB version are
• R goes down, The lower the variation decreases, there is a smaller increase. Need to evaluate more iterations so that residuals fall below as the convergence criterion.

Numerical Stability

• Iterations converged rapidly as the residual fell and was below the convergence criterion/ More complex problems converge more slowly. Condition which is determined by a stability analysis of the numerical scheme.
• Determined by performing stability analysis to the numerical scheme Numerical
• Want to take a small number of time-steps. time-step Δtmax after which it has a numerical scheme.
• The numerical errors grow exponentially in time
• There are two types of Explicit and implicit schemes
• The difference between the explicit and implicit can be illustrated using the wave equation. This gives the value for the grid points.
• Explicit Expression can be calculated which are easy to implement, and stable only C=(cΔt)/Delata x, in
• In implicit scheme , the derivative term is computed instead.
• This needs system of algebraic equations to calculate the time steps.
• Limits discussed as well, explicit schemes also need to have equation that have value 1.

Turbulence Modelling

• 2 different states of flow: laminar and turbulent flow
• Laminar flows are characterized by easily varying velocity.
• Flows rise and can dump perturbations of the flow for example (Reynolds number example. (pressure time relationship.