abstrac1coutte flow with heat transfer: numerical studies

8/13/2019 Abstrac1coutte flow with heat transfer: numerical studies

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Abstract

This project work report provides a full solution

of simplified Navier-Stokes equations for the

Incompressible Fully Developed Couette Flow

with heat transfer.

The known analytical solution to the problem is

compared with the numerical solution. For

discrete problem formulation implicit Crank-

Nicolson method is used. Finally, the system of

equation (TriDiagonal) is solved with Thomas

algorithm TDMA (TriDiagonal matrix algorithm).

Results are compared with the known analytical

solution to the problem.

1 Introduction

Figure-1 shows the main problem. There is a

viscous Fully Developed flow between two

parallel plates. Upper plate is moving in x-

direction with constant velocity(U). Lower one

is stationary. Temperature of the lower and

upper plates are T0and T1, respectively.

Figure-1: Schematic representation of Fully

Developed Couette Flow with plates at different

temperature problem

2 Fundamental Equations

Continuity Equation:

(mass conservation equation)

1)

X-Momentum equation:

2)

Y-Momentum Equation:

3)

Energy Equation:

4)

where, = viscous dissipation term

(5)

Assumptions:

1. Steady fully developed flow, which meansvelocity and temperature profiles do not change

in the flow direction.

2. Zero pressure gradient, zero body forces

3. Laminar, Incompressible flow

4. Constant properties of the fluid (, cp, , k)

with above assumptions,

(6)

Therefore, from equation(1), ; v= constant;Since v= constant and v=0 at bottom wall,

v=0 everywhere

X-momentum equation reduces to,

(7)

Y-momentum equation reduces to,

Hydrostatic pressure gradient would exist if

the body forces were present

Energy Equation will become as,

(8)


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Now non-dimensionalize momentum and

energy equations using the following

combinations:

; ;

(9)

Non-dimesionalized momentum equation

becomes:

(10)

Non-dimesionalized momentum equation

becomes:

(11)

where,

Br= Pr * Ec;

where, Pr= Prandtl Number =

Prandtl Number is fluid property,

and Ec = Eckert Number =

Boundary Conditions:

The solution to the momentum and energy

equation will give the following,

(12) (13)

When Ec Pr

The Non dimensional temperature profile

becomes linear. When Pr is of the order of

unity, small Ec implies dissipation effects

can be ignored.


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4 Numerical Solution

4.1 Velocity Profile

As we obtained in the last section,

X-momentum equation (10),

(14)

with boundary condition,

Finite-Difference Representation

In our solution of equation (14), we will use

Crank-Nicolson Scheme, so discrete

representation of that equation can be written

as:

(15)

(16)

with j=1,2,...,n (index 1 is at the bottom plate

and index n is at the upper plate)

and (points 0 and n+1 are imaginary points)

and

4.2 Temperature Profile

As we obtained in the last section,

Energy equation (11),

(17)

We know that , and therefore

Again using Crank-Nicolson Scheme, discrete

representation of equation(17) can be written

as:

(18)

(19)

with j=1,2,...,n (index 1 is at the bottom plate

and index n is at the upper plate)

and (points 0 and n+1 are imaginary points)

and

5 Solving The System of Linear Equations

The system of equations (16) and (18) can

either be solved by Gauss elimination method

or Thomas method (TDMA- TriDiagonal Matrix

Algorithm). I have used Thomas method.

The discretization equations can be written as

(20)

for i=1,2,3,...,N. Thus variable is related to theneighboring variables and . To accountfor the special form of the boundary-point

equations, let us set

and ,

so that the variables and will not haveany meaningful role to play.

Also

and

is given so we have

= given value of = given value of

Now for the matter of forward substitution, we

seek a relation

(21)after we have just obtained

(22)

Substituting equation (22) into equation (20)

leads to,

(23)

which can be rearranged to look like equation

(20). In other words, the coefficients and then stand for


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(24-a)

(24-b)These are recurrence relations.

Now set ,Now we are in position to start the back

substitution via equation (21).

Results

Dimensionless Velocity Profile

Dimensionless Temperature profile

Effect of Eckert number on Dimensionless

temperature Profile


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abstrac1coutte flow with heat transfer: numerical studies

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