ME - 733 Computational Fluid Mechanics Lecture 2drahmednagib.com/CFD_2018/CFD_Lecture_2.pdf ·...

ME - 733Computational Fluid Mechanics

Lecture 2

Dr./ Ahmed Nagib Elmekawy Oct 21, 2018

Leonhard Euler (1707-1783)

• In 1768, Leonhard Euler introduced the finite difference technique based on Taylor series expansion.

Lewis Fry Richardson (1881-1953)• In 1922, Lewis Fry Richardson developed the first numerical

weather prediction system.

• Division of space into grid cells and the finite difference approximations of Bjerknes's "primitive differential equations.”

• His own attempt to calculate weather for a single eight-hour period took six weeks and ended in failure.

• His model's enormous calculation requirements led Richardson to propose a solution he called the “forecast-factory.”

• The "factory" would have filled a vast stadium with 64,000 people.

• Each one, armed with a mechanical calculator, would perform part of the calculation.

• A leader in the center, using colored signal lights and telegraph communication, would coordinate the forecast.

1930s to 1950s

• Earliest numerical solution: for flow past a cylinder (1933).• A.Thom, ‘The Flow Past Circular Cylinders at Low Speeds’, Proc. Royal Society, A141, pp.

651-666, London, 1933

• Kawaguti obtained a solution for flow around a cylinder, in 1953 by using a mechanical desk calculator, working 20 hours per week for 18 months, citing: “a considerable amount of labour and endurance.”

• M. Kawaguti, ‘Numerical Solution of the NS Equations for the Flow Around a Circular Cylinder at Reynolds Number 40’, Journal of Phy. Soc. Japan, vol. 8, pp. 747-757, 1953.

1960s and 1970s• During the 1960s the theoretical division at Los Alamos contributed many numerical

methods that are still in use today, such as the following methods:• Particle-In-Cell (PIC).• Marker-and-Cell (MAC).• Vorticity-Streamfunction Methods.• Arbitrary Lagrangian-Eulerian (ALE).• k- turbulence model.

• During the 1970s a group working under D. Brian Spalding, at Imperial College, London, develop:• Parabolic flow codes (GENMIX).• Vorticity-Streamfunction based codes.• The SIMPLE algorithm and the TEACH code.• The form of the k- equations that are used today.• Upwind differencing.• ‘Eddy break-up’ and ‘presumed pdf’ combustion models.

• In 1980 Suhas V. Patankar publishes Numerical Heat Transfer and Fluid Flow, probably the most influential book on CFD to date.

1980s and 1990s• Previously, CFD was performed using academic, research

and in-house codes. When one wanted to perform a CFD calculation, one had to write a program.

• This is the period during which most commercial CFD codes originated that are available today:• Fluent (UK and US).• CFX (UK and Canada).• Fidap (US).• Polyflow (Belgium).• Phoenix (UK).• Star CD (UK).• Flow 3d (US).• ESI/CFDRC (US).• SCRYU (Japan).• and more, see www.cfdreview.com.

Navier-Stokes Equation Derivation

• Refer to

• Ch. 3 and Appendix A of• Jiyuan Tu, Computational Fluid Dynamics -A Practical Approach, Second Edition,

• Ch. 2• Wendt, Anderson, Computational Fluid Dynamics - An Introduction, 3rd edition 2009.

LAGRANGIAN AND EULERIAN DESCRIPTIONS

Kinematics: The study of motion.

Fluid kinematics: The study of how fluids flow and how to describe fluidmotion.

With a small number of objects, such

as billiard balls on a pool table,

individual objects can be tracked.

In the Lagrangian description, one

must keep track of the position and

velocity of individual particles.

There are two distinct ways to describe motion: Lagrangian and Eulerian

Lagrangian description: To follow the path of individual objects.

This method requires us to track the position and velocity of each individual

fluid parcel (fluid particle) and take to be a parcel of fixedidentity.

• A more common method is Eulerian description of fluid motion.

• In the Eulerian description of fluid flow, a finite volume called a flow domain

or control volume is defined, through which fluid flows in and out.

• Instead of tracking individual fluid particles, we define field variables,

functions of space and time, within the control volume.

• The field variable at a particular location at a particular time is the value of

the variable for whichever fluid particle happens to occupy that location at

that time.

• For example, the pressure field is a scalar field variable. We define the

velocity field as a vector field variable.

Collectively, these (and other) field variables define the flow field. The

velocity field can be expanded in Cartesian coordinates as

In the Eulerian description, one

defines field variables, such as

the pressure field and the

velocity field, at any location

and instant in time.

• In the Eulerian description we

don’t really care what happens to

individual fluid particles; rather we

are concerned with the pressure,

velocity, acceleration, etc., of

whichever fluid particle happens

to be at the location of interest at

the time of interest.

• While there are many occasions in

which the Lagrangian description

is useful, the Eulerian description

is often more convenient for fluid

mechanics applications.

• Experimental measurements are

generally more suited to the

Eulerian description.

CONSERVATION OF MASS—THE CONTINUITYEQUATION

To derive a differential

conservation equation, we

imagine shrinking a control

volume to infinitesimal size.

The net rate of change of mass within the

control volume is equal to the rate at

which mass flows into the control volume

minus the rate at which mass flows out of

the control volume.

Conservation of Mass: Alternative forms

• Use product rule on divergence term

kwjviuV

Conservation of Mass: Cylindrical coordinates

• There are many problems which are simpler to solve if the equations are written in cylindrical-polar coordinates

• Easiest way to convert from Cartesian is to use vector form and definition of divergence operator in cylindrical coordinates

Conservation of Mass: Cylindrical coordinates

Conservation of Mass: Special Cases

• Steady compressible flow

Cartesian

Cylindrical

0)()()(=

Conservation of Mass: Special Cases

• Incompressible flow

Cartesian

Cylindrical

= constant, and hence

kwjviuV

Conservation of Mass• In general, continuity equation cannot be used by

itself to solve for flow field, however it can be used to

1. Determine if a velocity field represents a flow.

2. Find missing velocity component

equation. continuity hesatisfy t torequired , w: Determine

flow ibleincompressan For

Example

zyzxyv

3 :Solution2

xzw +−−=

Conservation of MomentumTypes of forces:

1. Surface forces: include all forces acting on the boundaries of a medium though direct contact such as pressure, friction,…etc.

2. Body forces are developed without physical contact and distributed over the volume of the fluid such as gravitational and electromagnetic.

• The force F acting on A may be resolved into two components, one normal and the other tangential to the area.

If the differential fluid

element is a material

element, it moves with the

flow and Newton’s second

law applies directly.

Positive components of the stress

tensor in Cartesian coordinates on the

positive (right, top, and front) faces of

an infinitesimal rectangular control

volume. The blue dots indicate the

center of each face. Positive

components on the negative (left,

bottom, and back) faces are in the

opposite direction of those shown here.

Body Forces

Stresses (forces per unit area)

Double subscript notation for stresses.• First subscript refers to the surface• Second subscript refers to the direction• Use for normal stresses and for tangential stresses

Surface of constant xSurface of

constant -x

Complete Navier–Stokes equations

Newtonian versus Non-Newtonian Fluids

Rheological behavior of fluids—shear

stress as a function of shear strain rate.

Rheology: The study of the deformation of flowing fluids.

Newtonian fluids: Fluids for which the shear stress is linearly proportional to the shear strain rate.

Non-Newtonian fluids: Fluids for which the shear stress is not linearly related to the shear strain rate.

Viscoelastic: A fluid that returns (either fully or partially) to its original shape after the applied stress is released.

Some non-Newtonian fluids are called

shear thinning fluids or

pseudoplas t ic fluids, because the

more the fluid is sheared, the less

viscous it becomes.

Plastic fluids are those in which the

shear thinning effect is extreme.

In some fluids a finite stress called the

yield s t ress is required before the

fluid begins to flow at all; such fluids

are called Bingham plastic fluids.

Derivation of the Navier–Stokes Equation for Incompressible, Isothermal Flow

The incompressible flow

approximation implies constant

density, and the isothermal

approximation implies constant

viscosity.

Navier-Stokes Equations

)()]([)]([

22([)(

)()]([)].3

)]([)(

)()].3

22([)]([

)]([)(

x-momentum

y-momentum

z-momentum

Navier-Stokes Equations• For incompressible fluids, constant µ:

• Continuity equation: .V = 0

)]}[()][()]2[({

)]([)]([)].3

Navier-Stokes Equations• For incompressible flow with constant dynamic viscosity:

• x- momentum

• Y- momentum

• z-momentum

• In vector form, the three equations are given by:

2+−= Incompressible NSEwritten in vector form

Navier-Stokes Equations

)()]([)]([

22([)(

)()]([)].3

)]([)(

)()].3

22([)]([

)]([)(

x-momentum

y-momentum

z-momentum

Navier-Stokes Equations• For incompressible fluids, constant µ:

• Continuity equation: .V = 0

)]}[()][()]2[({

)]([)]([)].3

Navier-Stokes Equations• For incompressible flow with constant dynamic viscosity:

• x- momentum

• Y- momentum

• z-momentum

2+−= Incompressible NSEwritten in vector form

Navier-Stokes Equation• The Navier-Stokes equations for incompressible flow in

vector form:

• This results in a closed system of equations!• 4 equations (continuity and 3 momentum equations)• 4 unknowns (u, v, w, p)

• In addition to vector form, incompressible N-S equation can be written in several other forms including:• Cartesian coordinates• Cylindrical coordinates• Tensor notation

Incompressible NSEwritten in vector form

Euler Equations• For inviscid flow (µ = 0) the momentum equations are given by:

• x- momentum

• Y- momentum

• z-momentum

)()( ax

VD−=

Euler equationswritten in vector form

)()( by

)()( cz

Differential Analysis of Fluid Flow Problems

• Now that we have a set of governing partial differential equations, there are 2 problems we can solve• Calculate pressure (P) for a known velocity field

• Calculate velocity (U, V, W) and pressure (P) for known geometry, boundary conditions (BC), and initial conditions (IC)

• There are about 80 known exact solutions to the NSE

• Solutions can be classified by type or geometry, for example:1. Couette shear flows

2. Steady duct/pipe flows (Poisseulle flow)

Exact Solutions of the NSE

1. Set up the problem and geometry, identifying all relevant dimensions and parameters

2. List all appropriate assumptions, approximations, simplifications, and boundary conditions

3. Simplify the differential equations as much as possible

4. Integrate the equations

5. Apply BCs to solve for constants of integration

6. Verify results

• Boundary conditions are critical to exact, approximate, and computational solutions.▪ BC’s used in analytical solutions are

• No-slip boundary condition• Interface boundary condition

Procedure for solving continuity and NSE

Summary of Fluid Dynamic Equations in CFD Analysis

3D Compressible Navier–Stokes EquationsC.E. in conservative form

Complete Navier–Stokes equations in conservation form

3D Compressible Navier–Stokes EquationsC.E. in conservative form

0)()()(=

By expanding

3D Compressible Navier–Stokes EquationsMomentum Equations in conservative form

3D Compressible Navier–Stokes Equationswhen expanded using Stokes’ hypothesis (λ= - 2/3 μ) gives

3D Incompressible Navier–Stokes Equations

Continuity Equation

X-Momentum

Y-Momentum

Z-Momentum

𝜕𝑢

𝜕𝑥+𝜕𝑣

𝜕𝑦+𝜕𝑤

𝜕𝑧= 0

𝜌𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦+ 𝑤

𝜕𝑢

𝜕𝑧= 𝜌𝑔𝑥 −

𝜕𝑝

𝜕𝑥+ 𝜇

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2+𝜕2𝑢

𝜕𝑧2

𝜌𝜕𝑣

𝜕𝑡+ 𝑢

𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦+ 𝑤

𝜕𝑣

𝜕𝑧= 𝜌𝑔𝑦 −

𝜕𝑝

𝜕𝑦+ 𝜇

𝜕2𝑣

𝜕𝑥2+𝜕2𝑣

𝜕𝑦2+𝜕2𝑣

𝜕𝑧2

𝜌𝜕𝑤

𝜕𝑡+ 𝑢

𝜕𝑤

𝜕𝑥+ 𝑣

𝜕𝑤

𝜕𝑦+ 𝑤

𝜕𝑤

𝜕𝑧= 𝜌𝑔𝑧 −

𝜕𝑝

𝜕𝑧+ 𝜇

𝜕2𝑤

𝜕𝑥2+𝜕2𝑤

𝜕𝑦2+𝜕2𝑤

𝜕𝑧2

2D Incompressible Navier–Stokes Equations

Continuity Equation

X-Momentum

Y-Momentum

𝜕𝑢

𝜕𝑥+𝜕𝑣

𝜕𝑦= 0

𝜌𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 𝜌𝑔𝑥 −

𝜕𝑝

𝜕𝑥+ 𝜇

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2

𝜌𝜕𝑣

𝜕𝑡+ 𝑢

𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦= 𝜌𝑔𝑦 −

𝜕𝑝

𝜕𝑦++𝜇

𝜕2𝑣

𝜕𝑥2+𝜕2𝑣

𝜕𝑦2

Euler Equations

Continuity Equation

X-Momentum

Y-Momentum

Z-Momentum

𝜕𝑢

𝜕𝑥+𝜕𝑣

𝜕𝑦+𝜕𝑤

𝜕𝑧= 0

𝜌𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦+ 𝑤

𝜕𝑢

𝜕𝑧= 𝜌𝑔𝑥 −

𝜕𝑝

𝜕𝑥

𝜌𝜕𝑣

𝜕𝑡+ 𝑢

𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦+𝑤

𝜕𝑣

𝜕𝑧= 𝜌𝑔𝑦 −

𝜕𝑝

𝜕𝑦

𝜌𝜕𝑤

𝜕𝑡+ 𝑢

𝜕𝑤

𝜕𝑥+ 𝑣

𝜕𝑤

𝜕𝑦+ 𝑤

𝜕𝑤

𝜕𝑧= 𝜌𝑔𝑧 −

𝜕𝑝

𝜕𝑧

Poisson Equation

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2= 𝑓 𝑥, 𝑦

or 𝜕2𝜓

𝜕𝑥2+𝜕2𝜓

𝜕𝑦2= 𝑓 𝑥, 𝑦

Laplace Equation𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2= 0

or 𝜕2𝜓

𝜕𝑥2+𝜕2𝜓

𝜕𝑦2= 0

2D Viscous Burgers’ Equation (Convection)

2D Heat Equation (Diffusion)

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 𝜐

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2

𝜕𝑣

𝜕𝑡+ 𝑢

𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦= 𝜐

𝜕2𝑣

𝜕𝑥2+𝜕2𝑣

𝜕𝑦2

𝜕𝑢

𝜕𝑡= 𝜐

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2

2D Inviscid Burgers’ Equation (Convection)

2D Wave Equation (Linear Convection)

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 0

𝜕𝑣

𝜕𝑡+ 𝑢

𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦= 0

𝜕𝑢

𝜕𝑡+ 𝑐

𝜕𝑢

𝜕𝑥+ 𝑐

𝜕𝑢

𝜕𝑦= 0

1D Viscous Burgers’ Equation

1D Heat Equation (Diffusion)

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥= 𝜐

𝜕2𝑢

𝜕𝑥2

𝜕𝑢

𝜕𝑡= 𝜐

𝜕2𝑢

𝜕𝑥2

1D Inviscid Burgers’ Equation (Convection)

1D Wave Equation (Linear Convection)𝜕𝑢

𝜕𝑡+ 𝑐

𝜕𝑢

𝜕𝑥= 0

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥= 0

Basics of Finite Difference Formulations

Refer to

Ch. 2Hoffmann, A., Chiang, S., Computational Fluid Dynamics for Engineers, Vol. I, 4th ed., Engineering Education System, 2000.

Ch. 3, 4 and 5Pletcher, R. H., Tannehill, J. C., Anderson, D., Computational Fluid Mechanics and Heat Tranfer, 3rd ed., CRC Press, 2011.

Ch. 5Wendt, Anderson, Computational Fluid Dynamics - An Introduction, 3rd edition 2009.

• First step in obtaining a numerical solution is to discretize the geometric domain→

to define a numerical grid

• Each node has one unknown and needs one algebraic equation, which is a relation

between the variable value at that node and those at some of the neighboring

nodes.

• The approach is to replace each term of the PDE at the particular node by a finite-

difference approximation.

• Numbers of equations and unknowns must be equal

Discretization methods (Finite Difference)

• Numerical solutions can give answers at only discrete points in the domain, called grid points.

• If the PDEs are totally replaced by a system of algebraic equations which can be solved for the values of the flow-field variables at the discrete points only, in this sense, the original PDEs have been discretized.

• Moreover, this method of discretization is called the method of finite differences.

Discretization (Grid Generation)

• A partial derivative replaced with a suitable algebraic difference quotient is called finite difference.

• Most finite-difference representations of derivatives are based on Taylor’s series expansion.

• Taylor’s series expansion:Consider a continuous function of x, namely, f(x), with all derivatives defined at x. Then, the value of f at a location x+Δx can be estimated from a Taylor series expanded about point x, that is,

• In general, to obtain more accuracy, additional higher-order terms must be included.

Taylor’s series expansion

( ) ( ) ...)(!

fxfxxf

...)(!

1...)(

1)()()( 1

11 +−

+= +++++

iiiiiiii xxx

xxxfxfxf

)())(()()(

(xi+1-xi)= Δx step size (define first)

• The term, Rn, accounts for all terms from (n+1) to infinity, Truncation error.

)()()()(

xxfxfxf

𝑥𝑖+1 − 𝑥𝑖 = ∆𝑥 = ℎ

• Need to determine f n+1(x), to do this you need f'(x).

• If we knew f(x), there wouldn’t be any need to perform the

Taylor series expansion.

• However, R=O(Δxn+1), (n+1)th order, the order of truncation

error is Δxn+1.

• O(Δx), halving the step size will halve the error.

• O(Δx2), halving the step size will quarter the error.

Truncation Error

(1) Forward difference:

Neglecting higher-order terms,

we can get

Forward, Backward and Central Differences:

...)(!

)()()()()(

−+−

Solve for , we geti

Recall the Definition of a derivative:

𝜕𝑦

𝜕𝑥= lim

∆𝑥→0

𝑦2 − 𝑦1𝑥2 − 𝑥1

𝜕𝑦

𝜕𝑥= lim

∆𝑥→0

𝑦2 − 𝑦1∆𝑥

Finite Differences:

Recall the Definition of a derivative:𝜕𝑢

𝜕𝑥 𝑥𝑖= lim

∆𝑥→0

𝑢 𝑥𝑖+∆𝑥 −𝑢 𝑥𝑖

∆𝑥

Finite Differences:

(1) Backward difference:

(2) Forward difference:

(3) Central difference:

• This equation is known as the first forward difference approximation of of order (Δx).

• It is obvious that as the step size decreases, the error term is reduced and therefore the accuracy of the approximation is increased.

(1) Forward difference:( )

...)()(!

)()(!3

)()(!2

)()()(

−−−

−−

...)(!

...)(6

−−

Taylor series expansion:

Neglecting higher-order terms, we can get

Solve for , we get

(2) Backward difference

−+=+

−−++

−−

−+−

−−−−

)()1()(...)(

)()1(...

)()()()()(

)()(..)(!

)()1(...

)()()(

−−++

−−

• which represents the slope of the function at B using the values of the function at points A and B, as shown in Figure 2-2.

• Equation (2-6) is the first backward difference approximation

of of order (Δx).

(2) Backward difference

Figure 2-2. Illustration of grid points used in Equation (2-6).

• Adind (a)+(b) and neglecting higher-order terms, we can get

f iiiii +

−+−=

...)(!

...)(6

−−

)()1(...)(

)( 211 cxOx

f iii +

• which represents the slope of the function f at point B using the values of the function at points A and C, as shown in Figure 2-3.

• This representation of is known as the central difference approximation of order (Δx)2•

The higher-order term neglecting in Eqs. (a), (b), (c) constitute the truncation error.

Forward:

Backward:

Central:

Truncation error:

)()( 1 xOx

f iii +

)()( 1 xOx

f iii +

211 )(2

)( xOx

f iii +

if , then (a)+(b) becomes

* Central difference:

Second derivatives:

xxx ii == +1

2)( xO

f iiii +

( )...)(

)(...)(

+=+ in

iiiiix

( )...)(

)()1(...)(

)()()(

−=− in

iiiiix

HOTxOx

fxfff iiii ++

+=+ −+

11 )()()(2

If , then (b)-2(a) becomes

* Forward difference:

Second derivatives:

xxx ii == +1

f iiii +

( ) ( )HOT

fxffff iiiiii +

+−=− ++ )(

12 )()(2 xOx

fxfff iiii +

=+− ++

( )...)(

)(...)(

+=+ in

iiiiix

( )...)(

)2(...)(

)2(2)(

If , then (b)-2(a) becomes

* Backward difference:

Second derivatives:

xxx ii == +1

f iiii +

−−

( )...)(

)2(...)(

)2(2)(

−=− n

( ) ( )HOT

fxffff iiiiii +

+−=− −− )(

12 )()(2 xOx

fxfff iiii +

=+− −−

( )...)(

)()1(...)(

)()()(

−=− in

iiiiix

• By considering additional terms in the Taylor series expansions, a more accurate approximation of the derivatives is produced.

( )HOT

fff iiiii ...)(

1 )()(2

)( xOx

Substitute a forward difference expression for 2f/ x2 , i.e.,

f iiii +

121 )()()(

2)( xOxO

f iiiiii +

+−−

212 )(2

34)( xO

f iiii +

−+−=

More Accurate Approximations

gives:

ORa second-order accurate finite difference approximation

• For convenience, define the first forward difference fi+1 – fi as Δxf; and the first backward difference fi – fi-1 as ∇x f.

• In general, first order forward and backward differences can be

expressed as and

• Various central difference operators can be similarly defined. Some typical operators are:

Method of Operators

x ff = − )(1

x ff = −

ixixiiix fffff −=−= −+ 11

−=ii

ix fff

−+−+

+−=−−−=

iiiiiii

iixixxix

fffffff

11 2)( −+ +−=−= iiiixixixx ffffff

2 464)( −−++ +−+−= iiiiiixx ffffff

• Using the operators just defined, the approximations of the higher derivatives by forward, backward, and central differencing may be expressed as

Method of Operators

)( xOx

)()( xO

even)()(2

)( 222 nforxOx

ddnforxOx

o)()(2

Using operator method

Finite Difference Polynomial • Assume a second order polynomial

• Substituting with discrete points:

• Solving for A, B, and C

• Differentiating:

Finite Difference Equation• Consider the following equation:

Finite Difference Equation – Mixed Derivatives• Consider the 2D Taylor series expansion:

• Using i, j indices in replacement of x,y:

Finite Difference Equation – Mixed Derivatives• Similarly:

* Taylor series expansion:

* Central difference:

* Forward difference:

* Backward difference:

Mixed derivatives:

])(,)[(!2

)(),(),( 33

fxyxfyyxxf +

])(,)[())((4

221,11,11,11,1

f jijijiji

+−−=

−−−++−++

])(,)[())((

,,11,1,1

f jijijiji

+−−=

])(,)[())((

1,11,,1,

f jijijiji

+−−=

−−−−

• The finite difference approximations just discussed are used to replace the derivatives that appear in the PDEs.

• Consider an example involving time (t) and two spatial coordinates (x,y); i.e., the dependent variable f is f = f(t,x,y).

• A governing PDE of the form:

• where is assumed constant.

• It is required to approximate the PDE by a finite difference equation in a domain with equal grid spacing.

• The subscript indices i and j are used to represent the Cartesian coordinates x and y, and the superscript index n is used to represent time.

• It is specified that a first-order finite difference approximation in time and central differencing of second-order accuracy in space be used.

• The spatial grid spacings are Δx and Δy, whereas Δt designates the time step.

• The grid system is shown in Figure 2-6.

Finite Difference Equations

Computational grid system for the solution of Equation (2-26).

• Note that the value of f at time level n is known, and the value of f at time level n + 1 is to be evaluated.

• Therefore, Equation may be expressed at time level n or at time level n + 1.

• First, consider Eq. at time level n.

• For this case, a forward difference approximation which is first-order accurate is used.

• Hence,

• Therefore, the finite difference formulation of the partial differential equation (2-26) is:

• The second case, Eq. is evaluated at n + 1 time level.

• Therefore, a first-order backward difference approximation in time is employed, and the spatial approximations are at time level n+ 1.

• Hence, the finite difference formulation takes the form:

• The resulting finite difference equations, (2-27a) and (2-27b), are classified as explicit and implicit formulations, respectively.

• An obvious distinction between two finite difference equations is the number of unknowns appearing in each equation.

• Explicit equation involves only one unknown, fi,jn+1 , whereas Equation (2-

27b) involves five unknowns.

• Thus, the solution procedures based on explicit and implicit formulations

• are different.

Explicit and Implicit Formulations

• In the explicit formulation, only one unknown appears and may therefore be solved for directly at each grid point.

• In the implicit formulation, more than one unknown exists and therefore the finite difference equation must be written for all the spatial grid points at n+ 1 time level to provide the same number of equations as there are unknowns and solved simultaneously.

• Obviously, the solution of explicit formulation is simpler than the implicit equation.

• However, as will be seen shortly, implicit formulations are more stable than explicit formulations.

• Other differences between explicit and implicit formulations are discussed in future chapters.

• If the approximation of the original PDE was such that the truncation error was of the order [(Δt)2 , (Δx)2, (Δy)2 ], in which case the lowest term is of order two, then it would be classified as a second-order accurate method.

Explicit and Implicit Formulations

• Determine a backward difference approximation for f /x which is of the order (Δx)3

• Solution. Consider the Taylor series expansion,

• Substitute the backward difference approximations for 2 f I x2 and 3f I x3

Applications: Hoffman, Example 2.3

1 )()(!3

)())(( xO

fff ii +

−=−

• Substituting the expressions above into (2-33) produces:

Solution

• Using the Taylor series expansion, find a second-order forward difference approximation for f/x with unequally spaced grid points.

Hoffman, Example 2.4.

• . Given the function f(x) = ¼ x2 , compute the first derivative of f at x = 2 using forward and backward differencing of order (Δx). Compare the results with a central differencing of O(Δx)2

and the exact analytical value. Use a step size of Δx = 0.1. Repeat the computations for a step size of 0.4.

• Solution.

• From Equation (2-4), the forward difference approximation of order (Δx)

Hoffman, Example 2.6

• The backward difference approximation which is of order Δx

• The results obtained from backward and forward differencing deviate from the exact value when a larger step size is used. Selection of the step size is extremely important in numerical analysis.

Find a forward difference approximation of 0( Δx) for 4f/x4.

Hoffman, Example 2.1

• Therefore

Solution

• Determine the approximate forward difference representation for 3f /x3 which is of the order (Δx), given evenly spaced grid (a) Taylor series expansion

• (b) Forward difference recurrence formula

• Solution.

• (a) The Taylor series expansions of f(x+Δx), f(x+2Δx), and f(x+3Δx) about x are

Hoffman, Example 2.2.

Using Taylor series

1-D Wave equation

𝜕𝑢

𝜕𝑡+ 𝑐

𝜕𝑢

𝜕𝑥= 0

By applying Forward in time and central in space (FTCS)

𝑢𝑖𝑛+1 − 𝑢𝑖

∆𝑡+ 𝑐

𝑢𝑖+1𝑛 − 2𝑢𝑖

𝑛 + 𝑢𝑖−1𝑛

∆𝑥= 0

By rearranging

𝑢𝑖𝑛+1 = 𝑢𝑖

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖+1

𝑛 − 2𝑢𝑖𝑛 + 𝑢𝑖−1

1-D Wave equation

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖+1

Assume initial conditions of

𝑢 = 2@ 0.5 ≤ 𝑥 ≤ 1𝑢 = 1@ 𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒 𝑒𝑙𝑠𝑒

Assume Boundary conditions𝑢 = 1@ 𝑥 = 0, 2

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖+1

I.C.𝑢 = 2@ 0.5 ≤ 𝑥 ≤ 1

𝑢 = 1@ 𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒 𝑒𝑙𝑠𝑒

B.C.𝑢 = 1@ 𝑥 = 0, 2

1-D Wave equation

1-D Inviscid Burgers’ equation

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥= 0

∆𝑡+ 𝑢𝑖

𝑛 𝑢𝑖+1𝑛 − 2𝑢𝑖

∆𝑥= 0

By rearranging

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖

𝑛 × 𝑢𝑖+1𝑛 − 2𝑢𝑖

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖

Assume initial condition of

𝑢 = 2@ 0.5 ≤ 𝑥 ≤ 1𝑢 = 1@ 𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒 𝑒𝑙𝑠𝑒

Assume Boundary condition𝑢 = 1@ 𝑥 = 0, 2

𝑛 + 𝑐∆𝑡

∆𝑥× 𝑢𝑖

I.C.𝑢 = 2@ 0.5 ≤ 𝑥 ≤ 1

𝑢 = 1@ 𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒 𝑒𝑙𝑠𝑒

B.C.𝑢 = 1@ 𝑥 = 0, 2

1-D Diffusion equation

𝜕𝑢

𝜕𝑡= 𝜐

𝜕2𝑢

𝜕𝑦2

∆𝑡= 𝜐

∆𝑥 2

By rearranging

𝑛 + 𝜐∆𝑡

∆𝑥 2× 𝑢𝑖+1

1-D Diffusion equation𝜕𝑢

𝜕𝑡= 𝜐

𝜕2𝑢

𝜕𝑦2

∆𝑡= 𝜐

∆𝑥 2

By rearranging

𝑛 + 𝜐∆𝑡

∆𝑥 2× 𝑢𝑖+1

𝑛 + 𝑑 × 𝑢𝑖+1𝑛 − 2𝑢𝑖

Application to FTCS-Explicit scheme for the parabolic model equation

• Implicit scheme:𝑢𝑖𝑛+1 − 𝑢𝑖

∆𝑡= 𝜐

𝑢𝑖+1𝑛+1 − 2𝑢𝑖

𝑛+1 + 𝑢𝑖−1𝑛+1

∆𝑥 2

By rearranging

𝑛 + 𝜐∆𝑡

∆𝑥 2× 𝑢𝑖+1

𝑛+1 − 2𝑢𝑖𝑛+1 + 𝑢𝑖−1

𝑛+1

Let r = 𝜐∆𝑡

∆𝑥 2

𝑢𝑖𝑛+1 += 𝑢𝑖

𝑛 + 𝑟 × −2𝑢𝑖𝑛+1 + 𝑢𝑖−1

𝑛+1

− 𝑟 𝑢𝑖+1𝑛+1 + 1 + 2𝑟 𝑢𝑖

𝑛+1 − 𝑟𝑢𝑖−1𝑛+1 = 𝑢𝑖

Compare the previous Eq. with the following(where 𝑖 = 1: m)𝑎𝑖 𝑢𝑖+1

𝑛+1+ 𝑑𝑖 𝑢𝑖𝑛+1+ 𝑏𝑖 𝑢𝑖−1

𝑛+1= 𝑐𝑖

where m: number of nodes in y direction,

𝑎𝑖 = −𝑟, 𝑑𝑖 = 1 + 2𝑟, 𝑏𝑖 = −𝑟, 𝑐𝑖 = 𝑢𝑖𝑛

• Implicit scheme:We will get that

Rewriting previous Eqns. In matrix form

• Implicit scheme:

Parabolic PDEs:

• Richardson method:

– The scheme is second order time and space

– Unconditionally unstable (has no practical value)

Parabolic PDEs:

• DuFort-Frankel method

– The scheme is second order in time and second order in space

– For stability considerations the term in the diffusion term isreplaced with an average between and

– The resulting equation is explicit formulation

– The scheme is unconditionally stable

Parabolic PDEs: Implicit Methods• Crank-Nicolson method

– The scheme is second order time and space

– The diffusion term is replaced by a time average value of thesecond order central space difference term

– The second order in time is unclear, but if we thought of as asummation of two-step computations:

1. Explicit step

2. Implicit step

Programming Assignment 1Consider a fluid bounded by two parallel plates extended to infinity such that no end effects are encountered. The walls and the fluid are initially at rest. Now, the lower wall is suddenly accelerated in the x-direction. The Navier-Stokes equations for this problem may be expressed as:

𝜕𝑢

𝜕𝑡= 𝜐

𝜕2𝑢

𝜕𝑦2

It is required to compute 𝑢 𝑡, 𝑦 .

Programming Assignment 1Assume initial conditions of

𝑢 = 𝑢𝑜 @ 𝑦 = 0𝑢 = 0@ 0 < 𝑦 ≤ ℎ

where h is the distance between the two plates and equals 40 mm.

Assume Boundary conditions𝑢 = 𝑢𝑜 @ 𝑦 = 0𝑢 = 0@ 𝑦 = ℎ

Take 𝜐=0.000217 m2/s, 𝑢𝑜 =40 m/s, max time of 1.08 sec. Assume 40 nodes in y direction

Programming Assignment 1Apply FTCS scheme.

Calculate and plot the velocity distribution by usingMatlab by using the following time steps:

❑ dt = 0.002 sec

❑ dt = 0.00232 sec

❑ dt = 0.003 sec

Bonus points will be given to the student who

• Creates a video of the development of the flow speed with time.

ME - 733 Computational Fluid Mechanics Lecture 2drahmednagib.com/CFD_2018/CFD_Lecture_2.pdf ·...

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