Imaginative Consultancy in Computational Fluid Dynamics
Computational Fluid Dynamics Recipes - Volume
Table of Contents
Preface
List of Examples
List of Codes
I - Basics Matters
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1 - Vectors and Tensors
1.1 Scalar and Vector functions
1.1.1 Scalar functions
1.1.2 Vector functions
1.2 Vector multiplication
1.2.1 Dot product
1.2.2 Cross product
1.2.3 Dyadic product
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1.3 Differentiation of a vector
1.3.1 Derivative of a vector function
1.3.2 Vector differential operator ∇
1.3.3 Gradient of a scaler field
1.3.4 Divergence of a vector field
1.3.5 Curl of a vector
1.4 Integral Theorems
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1.4.1 Gauss' theorem
1.4.2 Stokes' theorem
1.4.3 Leibniz's theorem
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1.5 Vector Operators in other coordinates
1.5.1 Gradient of a Scalar Field
1.5.2 Divergence of a Vector Field
1.5.3 Curl of a Vector Field
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2 - Matrices and Matrix Manipulations
2.1 Multiplication of Matrices
2.2 Some Special Matrix Forms
2.2.1 The Null Matrix
2.2.2 Square and Diagonal matrices
2.2.3 The Identity or Unit Matrix
2.2.4 Triangular Matrices
2.2.5 Tridiagonal Matrix
2.2.6 Banded Matrices
2.2.7 Diagonally Dominant Matrices
2.3 Norms
2.3.1 Important Norms
2.4 Transpose of a matrix
2.5 Symmetry in Matrices
2.5.1 Skew Symmetric Matrices
2.5.2 Symmetric and Skew Symmetric Decomposition
2.6 The matrix inverse and its significance
2.7 Condition of a matrix
2.7.1 Condition number
3 - Linear Solvers
3.1 Iteration Methods
3.1.1 Jacobi and Gauss-Seidel Methods
3.1.2 The sequence of the Jacobi and Gauss-Seidel methods
3.1.3 Successive-Over-Relaxation Method (SOR)
3.2 Convergence of iterative method
3.3 Example: Solving a System Using SOR
4 - Classification of Partial Differential Equations
4.1 Elliptic Partial Differential Equations
4.2 Parabolic Partial Differential Equations
4.3 Hyperbolic Partial Differential Equations
4.4 The Behavior of the fluid flow equation
5 - Discretization Issues
5.1 Discretization Methods
5.1.1 Finite Difference Methods
5.1.2 Finite Volume Methods
5.2 Grids or Meshes
5.2.1 Body-fitted regular grids
5.3 Properties of Numerical Methods
5.3.1 Accuracy
5.3.2 Consistency
5.3.3 Stability
5.3.4 Convergence
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II - Computational methods for steady state diffusion problems
6 - Scalar Transport Equation
6.1 Conservation of a Scalar Property
6.2 Diffusion dominated problem
7 Steady Diffusion Dominated Problems
7.1 Integral Equations
7.2 One Dimensional Problem
7.2.1 The Integral Equation
7.2.2 The Grid
7.2.3 Discretization
7.2.4 Solving a System of Linear Algebraic Equations
7.3 Multi-Dimensional Problem
7.3.1 The Integral Equation
7.3.2 The Grid
7.3.3 Discretization
7.4 Boundary Conditions
7.4.1 Dirichlet Boundary Conditions
7.4.2 Neumann Boundary Conditions
7.4.3 Mixed Boundary Conditions
8 Steady State Diffusion Examples
8.1 One-dimensional Heat Conduction
8.1.1 Integral equations
8.1.2 The Grid
8.1.3 Discretization
8.1.4 System Solver
8.1.5 The code
8.2 One-Dimensional Heat Conduction: Circular Fin
8.2.1 Integral Equation
8.2.2 The Grid
8.2.3 Discretization
8.2.4 System Solver
8.2.5 The Code
8.3 Two-Dimensional Heat Conduction
8.3.1 Integral Equations
8.3.2 The Grid
8.3.3 Discretization
8.3.4 System Solver
8.3.5 The code
8.4 Two Dimensional Conduction - with a Sink or a Source
8.5 Heat Conduction in a Circular Rod
8.5.1 Integral Equations
8.5.2 The Grid
8.5.3 Discretization
8.5.4 System Solver
8.5.5 Results
8.5.6 The Code
8.6 Two-Dimensional Plate With a Rectangular Hole
8.6.1 Integral Equation
8.6.2 The Grid
8.6.3 Discretization
8.6.4 Results
8.6.5 The Code
III - Fluid Dynamics
9 - Conservation Laws of Fluid Dynamics
9.1 Conservation of Mass: Continuity Equation
9.2 Conservation of Linear Momentum
9.2.1 Stress Tensor - σ
9.2.2 Newtonian Fluids
9.3 Conservation of Energy
9.3.1 Rate of change of energy
9.3.2 Rate of Work
9.3.3 Energy Transfer
9.4 Conservation Laws In The Cartesian coordinates
9.5 Some General Observations
9.5.1 Diffusion
9.5.2 Convection, Advection, and Fluxes
9.6 Conservation and Non-Conservation Forms
9.7 Non-Dimensional Groups
10 Viscous Incompressible Flows
10.1 Continuity Equation
10.2 Momentum Equation
10.3 Energy Equation
10.4 Summary
10.5 Observations about incompressible flow assumption
11 Vorticity-Stream Function Formulation
11.1 Vorticity Formulation in Cartesian Coordinates
11.2 Vorticity Formulation in Cylindrical Coordinates
11.2.1 Polar Coordinates
IV - Computational methods for steady state creeping flows
12 Diffusion Problems - Revisited
12.1 Non-dimensional Navier-Stokes Equations
12.2 Diffusion dominated flows
13 Steady State Creeping Flow Examples
13.1 Two-dimensional Vorticity-Stream Function formulation
13.2 Two-Dimensional Lid-Driven Cavity Creeping Flow
13.2.1 Simplified Staggered Grid
13.2.2 The Integral Equations
13.2.3 Boundary condition
13.2.4 Discretization
13.2.5 Results
13.2.6 The Code
13.3 Creeping Flow past a square block on a plate
13.3.1 The Grid
13.3.2 Integral Equations
13.3.3 Discretization
13.3.4 Results
13.3.5 The Code
V - Convection & Diffusion of a scalar
14 Steady State Convection - Diffusion of a Scalar
14.1 Scalar Transport Equation
14.2 One-Dimensional Problem
14.2.1 The Grid
14.2.2 Discretization
14.3 Central Differencing
14.3.1 The Grid
14.3.2 Discretization
14.3.3 Results
15 Upwind Method
15.1 One-Dimensional Upwind Differencing
15.1.1 The Grid
15.1.2 Discretization
15.1.3 Results
15.1.4 The Code
15.2 Extension to multi dimensions
15.2.1 Two-Dimensional Case
15.2.2 Three-dimensional Case
15.3 General Formulation
15.4 Boundary Conditions
15.4.1 Dirichlet Boundary Conditions
15.4.2 Neumann Boundary Conditions
16 Properties of Discretization Methods
16.1 Properties of discretization methods
16.1.1 Conservativeness: Consistency at control volume faces
16.1.2 Boundedness
16.1.3 Transportiveness
16.1.4 Negative-slope linearization of the source term
16.1.5 Sum of the neighbor coefficients
16.2 Assessment of the central difference scheme
16.2.1 Conservativeness
16.2.2 Boundedness
16.2.3 Transportiveness
16.2.4 Accuracy
16.3 Assessment of the first-order upwind method
16.3.1 Conservativeness
16.3.2 Boundedness
16.3.3 Transportiveness
16.3.4 Accuracy
16.4 Numerical Diffusion and Dispersion
16.4.1 Upwind Method
16.4.2 Central Differencing Method
17 Steady Upwind Method Examples
17.1 Lid-Driven Cavity Flow: Upwind Method
17.1.1 The Grid
17.1.2 The Integral Equations
17.1.3 Boundary condition
17.1.4 Discretization
17.1.5 Results
17.1.6 The Code
17.2 Flow Over a Backward Facing Step: Upwind Method
17.2.1 The Grid
17.2.2 The Integral Equations
17.2.3 Boundary conditions
17.2.4 Discretization
17.2.5 Results
17.2.6 The Code
17.3 Flow past a square block on a plate: Upwind method
17.3.1 The Grid
17.3.2 The Integral Equations
17.3.3 Boundary condition
17.3.4 Discretization
17.3.5 Results
17.3.6 The Code
18 Other first-order Schemes
18.1 Hybrid Scheme
18.1.1 One-dimensional Hybrid Formulation
18.2 One-Dimensional Hybrid Example
18.2.1 The Grid
18.2.2 Discretization
18.2.3 Results
18.2.4 The Code
18.3 Hybrid Method: Multi-Dimensional Formulation
18.3.1 Two-Dimensional Case
18.3.2 Three-dimensional Case
18.4 Boundary Conditions
18.4.1 Dirichlet Boundary Conditions
18.4.2 Neumann Boundary Conditions
18.5 CONDIF Scheme
18.5.1 Multi-dimensional CONDIF Formulation
19 Hybrid & CONDIF Examples
19.1 Lid-Driven Cavity Flow - Hybrid Method
19.1.1 Results
19.1.2 The Code
19.2 Lid-Driven Cavity Flow: CONDIF Method
19.2.1 Results
19.2.2 The Code
19.3 Flow past a square block on a plate: Hybrid method
19.3.1 Results
19.3.2 The Code
19.4 Flow past a square block on a plate: CONDIF Method
19.4.1 Results
20 Higher-order Formulations
20.1 Second-Order Upwind Scheme
20.1.1 Fromm Scheme
20.1.2 Beam and Warming Scheme
20.2 Third-order Upwind QUICK Scheme
20.3 One-Dimensional QUICK Example
20.3.1 The Grid
20.3.2 Discretization
20.3.3 Results
20.3.4 The Code,
20.4 Properties of QUICK scheme
20.5 Multi-Dimensional Formulation
20.5.1 Two-Dimensional Formulation
20.5.2 Three-dimensional Formulation
20.6 Boundary Conditions
20.6.1 Dirichlet Boundary conditions
20.7 Stability,
21 QUICK Scheme Examples
21.1 Lid-Driven Cavity: QUICK Scheme
21.1.1 The Grid,
21.1.2 The Integral Equations
21.1.3 Boundary conditions
21.1.4 Discretization
21.1.5 Results
21.1.6 The Code
21.2 Flow Over a Backward Facing Step
21.2.1 The Grid
21.2.2 Boundary Conditions
21.2.3 Discretization
21.2.4 Vorticity Equation
21.2.5 Stream Function
21.2.6 Velocities
21.2.7 Results
VI - Primitive variables formulation
22 Primitive Variables and Staggered Grids
22.1 Staggered Grid
22.2 Momentum Equations on a Staggered Grid
22.2.1 u-Momentum Equation
22.2.2 v-Momentum Equation
22.3 Non-Dimensional Equations on the Staggered Grid
22.3.1 The Non-Dimensional u-Momentum Equation
22.3.2 The Non-Dimensional v-Momentum Equation
22.4 The Solution Strategy: Pressure Correction Method
23 SIMPLE Family & PISO Algorithms
23.1 The Pressure and Velocity Corrections
23.1.1 Velocity Correction Formula
23.1.2 Pressure Correction Equation
23.2 SIMPLE Algorithm
23.2.1 Under-relaxation
23.2.2 The Sequence of the SIMPLE Algorithm
23.3 Boundary Conditions
23.4 SIMPLER Algorithm
23.4.1 The Sequence of the SIMPLER Algorithm
23.5 SIMPLEC Algorithm
23.5.1 The sequence of the SIMPLEC algorithm
23.6 General observations on SIMPLE Family Algorithms
23.7 PISO Algorithm
23.7.1 The Sequence of the PISO Algorithm for Steady State Problems
24 SIMPLE Algorithm Examples
24.1 One-Dimensional Nozzle Flow: SIMPLE-upwind schemes
24.1.1 The exact solution
24.1.2 The Grid
24.1.3 Governing Equations
24.1.4 Discretization
24.1.5 Convergence
24.1.6 The Results
24.1.7 The Code
24.2 Lid-Driven Cavity Flow: SIMPLE-upwind
24.2.1 The Grid
24.2.2 Boundary condition
24.2.3 Integral Equations
24.2.4 Discretization
24.2.5 Results
24.2.6 The Code
24.3 One-Dimensional Nozzle Flow: SIMPLER- upwind schemes
24.3.1 Discretization
24.3.2 The Results
24.3.3 The Code
VII Unsteady Convection & Diffusion
25 Unsteady Diffusion Dominated Problems
25.1 Unsteady Diffusion Problems
25.2 Integral Equations
25.3 The Explicit Scheme
25.3.1 One-Dimensional Transient Heat Conduction - Explicit Scheme
25.3.2 The Analytical Solution
25.3.3 Integral Equations
25.3.4 Explicit discretization
25.3.5 Results
25.3.6 The Code
25.3.7 Properties of the explicit scheme
25.4 The Fully-Implicit Scheme
25.4.1 One-Dimensional Transient Heat Conduction - Fully Implicit Scheme
25.4.2 Results
25.4.3 The Code
25.4.4 Properties of the fully-implicit scheme
25.5 The Crank-Nicholson Scheme
25.5.1 One-Dimensional Transient Heat Conduction- Crank-Nicholson Scheme
25.5.2 Results
25.5.3 The Code
25.5.4 Properties of the Crank-Nicholson scheme
25.6 Unsteady Lid-Driven Cavity creeping Flow
25.6.1 The Vorticity Equation
25.6.2 The Stream Function and the Velocities
25.6.3 Results
25.6.4 The Code
26 Unsteady Scalar Transport Equation
26.1 The Reynolds Transport Equation
26.2 Fully Implicit Scheme
26.2.1 Upwind method
26.2.2 Hybrid method
26.3 Unsteady Flow Over a Backward Facing Step
26.3.1 The solution procedure
26.3.2 Discretization
26.3.3 Results
26.3.4 The Code
27 Unsteady Primitive Variable Formulation: SIMPLE
27.1 SIMPLE - upwind and fully-implicit discretizations
27.2 Unsteady formulation and the under-relaxation
27.3 Steady state problems with unsteady formulation
27.4 Unsteady Cavity Flow: SIMPLE - upwind - implicit
27.4.1 Results
27.4.2 The Code
References
Index