asimov studies

Introduction

fluid flows are governed by partial differential equations which represent conservation laws for the mass, momentum, and energy.

Presented as: 1- mass is conserved, 2- Newton's Second Law (Force = mass x acceleration), 3- Energy is conserved

CFD introduces computational methods that replace PDE (Partial Differential Equations) by a set of algebraic equations that solves the flow by:

Algebraic equations (PDE)

Numerical  methods

The flow under study are characterized by the table defining them as:

Compressible

a flow is considered compressible if the density of the fluid changes with respect to pressure

Incompressible

a flow is considered incompressible if the density of the fluid does not change with respect to pressure

Steady

a flow is considered steady if the velocity does not change with time

Transient

a flow is considered transient if the velocity does change with time

Viscous

a viscous flow is characterized by considerable viscous effects

Inviscid

an inviscid flow is characterized by no viscous effects

One phase

a one phase flow is characterized by one substance solid or liquid or gaz

Multi phase

Multiphase flow is characterized by multiple substances (solid, liquid, gaz)

Before any solution is available, the following steps have to be implemented and checked:

Pre Processing

the aim of a computational method is to solve a differential equation of the variable under study. the numerical method treats its basic unknowns the values of the dependent variable at a defnite number of locations called nodes, on a Grid. at each node, a discrete value of the variable is available, thus the continuous solution given by an analytical solution is discreticized into a set of values.

the discretization method will replace the differential equations by a set of algebraic equations that describes how the varibale does change between the grid points.

 

1-      Mesh

The mesh is the discretization of the control volume into grids, the mesh can be formed of the following Elements:

1-      Polygonial2-      Tetrahedral 3-      Structured 4-      Unstructured 5-      Quadrilateral 6-      triangular

look at mesh size as refined or coarse.

2-      Space discretization

Space Discretization can be of one of the following methods:

Finite difference Method

Finite Element Method

Finite Volume Method

(all can be of High vs low order approximations)

3-      Time discretization

Explicit vs Implicit

an explicit solution is presented by the variable only at the left side of the solution, and only to the first power.

an implicit solution is any solution that is not explicit.

Local time stepping (coarse or fine time step)

4-      Stopping criteria

Checking residuals

5-      Iterative Convergence

Effects on Round off error.

6- Initial and boundary conditions

Define the initial and boundary conditions

boundary conditions: heat dissipation, air flow, pressure, temperature.

boundary condition walls: adiabatic or insulated.

7- Assumptions

natural convection: h = 5 (w/m2 k).

forced convection: h > 10 (w/m2 k).

Post Processing

Calculation of values: Lift, Drag, Velocity, Pressure, Temperature, Vibration Modes.

1D data: point values

2D data: contours

3D data: isosurface, isovolumes

PDE (PARTIAL DIFFERENTIAL EQUATIONS)

Working on CFD requires an understanding of the partial differential equations (PDE) under study.

Hyperbolic Differential equation

the Hyperbolic Differential equation is a second order differential equation with the following form:

with

A2-4BC > 0

Parabolic Differential equation

the Parabolic differential equation is a second order differential equation with the following form:

with

A2-4BC = 0

Elliptic Differential equation

the Elliptic differential equation is a second order differential equation with the following form:

with

A2-4BC < 0

 

FEA BASICS

Cartesian Coordinate

the cartesian coordinate represents three dimensional axes, each node is represented by a numerical coordinate on the three axes

Vector

a vector is represented by a magnitude and direction.

a vector is formed by combining two nodes on the cartesian coordinate system

example

the magnitude of a vector is defined as below

Unit Vector (of a vector)

Substantial Derivative

Substantial derivative represents the instantaneous time rate change of the variable as it moves through a an element. In this example, it represents the instantaneous change of density of a fluid as it moves throught element 1.

Vector Differential Operator

vector operator represents the time rate of change of the unit element per unit volume

Gradient

gradient is mainly used with scalar rather than vector operation, the gradient of a scalar is given below

Divergence

the divergence of the vector is:

Divergence theorem

the divergence theorem " Gauss's divergence theorem " can be written in vector notation

the theorem specifies that any change within the volume is equal to the same amount flowing in and out of the outside surface.

Laplacian Operator

Curl

Kronecker Delta

δij =1 if i = j
δij =0 if i ≠ j

MATH BACKGROUND

deteminant of a matrix

the determinant of a matrix A is defined as det [A] or |A| or |det A| and is computed as follows in the example:

Example:

vector product on a matrix

the vector product of two vectors a and b is represented by:

or in matrix format

Example: