Final Project Report

AME 535a

Fall 2012

Rohith Jayasimha Rao

USC ID: 7624-4415-70

AME Department

University of Southern California

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Contents

• Problem definition

• Mesh and Time Step definition

• Method of Solution

• Results and Plots

• Discussion of results

• Some additional plots

• Scope for further work

• Listing of the code and a description of the various subroutines used.

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Problem Definition:

This project report details a numerical solution method to a steady, viscous entrance flow into a two-

dimensional plane channel. The channel walls are separated by a constant distance, 2h. The flow enters at

(x=0), with a uniform velocity (u0=1). The two-dimensional Navier-Stokes equations form the basis for

the solution, which are represented as follows in Cartesian coordinates:

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The non-dimensional versions of the two equations are calculated and used in further analysis. The

simplifications of these equations leading to the specific form in which they are implemented in the

project are presented separately in hand-written form and attached to this project report.

As such, a constant mean pressure gradient is assumed. The value of this constant is found out to be equal

to (3/Re) i.e.

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��� �|�|� Where |C|=3/Re; Where Re= Reynolds Number.

The equations in their final form as they are implemented in the problem are in the following form:

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And a similar equation for the v-velocity component can be written as:

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The boundary conditions applied in this problem are as follows:

• No-slip boundary conditions at the channel walls.

• No through flow conditions at the walls.

• A uniform streamwise velocity at the inlet, with a magnitude of uo=1.

• Vertical velocity component is zero at the outflow, i.e., v (xmax,y)=0.

• A Neumann boundary condition for the streamwise velocity at the outflow that states that: ��������� ������

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For such a flow, an analytical flow can be obtained for the steady state flow at the exit of the channel.

This solution has the following form:

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The vertical profile of the horizontal velocity, u, hence looks like a parabola with the peak value being at

1.5 in our case. The numerically obtained solutions have been compared to this analytical solution in

order to test the accuracy of the numerical schemes employed.

The exact solution at the outflow is depicted in the plot below, which shows the vertical velocity profile

of the streamwise velocity component:

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Mesh Definition:

The mesh is constructed such that there are nx points in the x-direction and ny points in the y-direction.

The nodal points are numbered starting from ‘1’ in either direction. The nodal points form a grid and the

flow quantities are calculated at each of these grid points.

The distance between any two neighboring nodal points in the x-direction is defined by the quantity ∆x,

and the distance between any two neighboring points in the y-direction is defined by the quantity ∆y. The

distance between the points does not vary from one pair of neighboring points to the other in any given

direction. Hence, the grid is uniform in the x-direction and uniform in the y-direction. However, the

number of points ‘nx’ can vary from the number of points ‘ny’ in the y-direction. This difference is

especially protracted in the case of the particular problem being examined in this report. This is due to the

fact that the height of the channel is always fixed, with the y-coordinate going from y=-1 to y=1. The

length of the channel, xmax, however can be varied until convergence is achieved for a particular value of

Reynolds number.

Time Step Definition:

The time step is defined on the basis of the Courant-Friedrichs-Lewy (CFL) number. The condition for

stability here is that the CFL number, C should be less than 1. Hence the time-step ∆t, is set so that this

stability condition is enforced. The relation relating the time step to the CFL number is as follows:

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Where u is the constant inlet velocity.

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Method of SolutionA pseudo-transient, fractional step method, alternatively known as the ‘time-splitting’ method is used.

The steps involved in the solution are:

1. The nonlinear step that takes into account the convective terms of the Navier- Stokes equations.

This step results in an intermediate value of the velocity components denoted by u* and v

*. At the

first time step, the initial conditions are fed as inputs to this step.

2. The pressure step that takes into account the pressure term in the Navier-Stokes equation. In the

form of the equation that we are solving in this project, this term involves the perturbation

pressure p(x,y). This step results in an intermediate value of the velocity components denoted by

u**

and v**

.

3. Two viscous steps. The two viscous steps take into account the effects of viscosity. After solving

both the viscous steps, a final value of both the velocity components is arrived at. These are

denoted as un+1

and vn+1

. These are the values are the values of the velocity components after one

complete time step.

4. At the end of one complete time step, the final values of velocity are fed back in to the non-linear

step and the entire process is repeated until convergence is achieved.

5. The convergence is tested on the basis of the values of the streamwise velocity matrix. A preset

tolerance level is set, which is the minimum variation between the streamwise velocity values

between two consecutive time steps, at all the grid points, which is to be achieved in order to

achieve a convergent solution.

6. As soon as convergence is detected, the iterations stop and an rms error is calculated at the

outflow using the analytical solution as the benchmark.

7. Also, the flow rate is calculated at every point along the length of the channel by numerically

integrating the velocity values along the height of the channel. This computed value is compared

to the constant inlet flow rate and an rms error is again computed.

Each of the steps is explained in detail along with a presentation of relevant formulae and a brief

discussion of the discretization procedures followed to arrive at the said formulae.

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The Non-Linear Step

Consider the non-linear part of the Navier-Stokes equations: ���� � ������ � ��

Where

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And,

�� � �� ���� � � ����The spatial derivatives that constitute the terms Fu and Fv are discretized using a standard central

difference scheme. The time advancement is done using the Adams-Bashforth method, i.e.

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And, �� � �

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As is evident, the Adams-Bashforth method is a three-level system that requires information at two

previous time levels as input. At the first step, as we have the values of velocity for only one previous

time step, a different two-level scheme must be employed to start the solution process. The method

chosen for this is the Euler time discretization. The formula for the same can be expressed as:

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Here the level n=1, is taken as the initial condition. Consequently, the solution after the first time step is

the solution at time level n=2.

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Spatial discretization of the velocity terms:

Application of a standard central difference scheme, gives the following discretized versions of the spatial

terms Fu and Fv:

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And,

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�� Substituting these values into the equation for the Adams Bashforth method yields the following

equations:

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And,

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�� $The algorithm that arises from the simplification of these equations has been implemented in the

subroutine Convec.m . The algorithm is as follows:

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And,

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The same substitution in the Euler method yields the following algorithm:

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And,

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The above algorithms are applied to the internal points only. However, we must include the outflow

boundary into our computational domain and solve for the streamwise velocity values in order to enforce

the Neumann boundary condition at the outflow boundary that was stated earlier. This is done as follows:

At the boundary the Neumann boundary condition is given by: ��

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This is discretized using a central difference scheme. In order to do so, we must introduce a fictitious

point beyond the computational domain at the point (xmax+dx). In terms of nodal values, this can be

viewed as the introduction of a fictitious node (nx+1). Such a discretization yields:

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or, ����� � ����

Hence, all the terms that are related to the streamwise velocity at the (nx+1)th node are equated to the

respective values at the (nx-1)th node. Hence, a different algorithm takes shape that is implemented only at

the outflow boundary.

The values of u*and v

* that are obtained from this step are passed onto the subroutine that performs the

calculations relating to the pressure step.

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The Pressure Step (Press.m or Press0.m):

The pressure step is calculated one of the two routines Press.m or Press0.m. The difference between the

two subroutines is the way in which the right hand side of the Poisson equation for pressure is defined.

Although the difference is subtle, it was found to produce varying results with the technique implemented

in Press.m (here onwards referred to as the Press technique) proving to be unstable at Re=100, whereas

the technique implemented in Press0.m (here onwards referred to as the Press0 technique) fared well at

that high value of Re. Results based on both techniques have been discussed and compared in subsequent

sections of the report. The partial differential equations that form the basis of both the techniques are

however the same and a general discussion of the same follows.

The pressure step solves the pressure term in the Navier-Stokes equation, temporarily neglecting all other

terms. The time discretization of the step is as follows:

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�� � � The value of the perturbation pressure at all the grid points is in turn obtained from a Poisson equation for

p:

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Subjected to the boundary conditions: ��� � %�� � ��&%������ � %�� � ���� � ��

Once the values of p are known, the values of u**

and v**

are calculated using the following equations:

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Discretization of the left hand side of the Poisson equation for Pressure:

The spatial pressure derivatives are discretized using a central difference scheme as follows:

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The above discretized versions of the derivatives are substituted in the Poisson equation to obtain the

following expression:

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The right hand side of the above equation is known (the discretization is discussed in the next section).

Thus, we have a set of (nx×ny) linear equations in (nx×ny) unknowns. This is solved using the LU

decomposition with the help of the subroutines FACT and SOLVE.

The Neumann boundary conditions at all the boundaries mean that we must solve for the pressure terms at

all the nodal points, i.e. at (nx × ny) points. This is achieved by introducing fictitious nodes at all the

boundaries akin to introducing an entire fictional boundary that surrounds the computational domain. The

Neumann boundary conditions are discretized using a second order, central difference scheme, so that the

following equalities hold good:

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The above substitutions are made when writing the equations for the boundary points. The scheme is thus

implemented at all the points in the computational domain.

However, this method has a tendency to produce unusually large values of pressure. The values of

pressure are hence controlled by setting the value of p(nx,ny)=1. Thus, we now have one variable less to

solve for and we end up solving only for [(nx×ny)-1] unknowns.

Discretization of the right hand side of the Poisson equation for pressure:

The right hand side of the equation is discretized using two methods:

The Press Technique:

The Press technique involves the usage of backward difference formulae to discretize the spatial

derivatives of the velocity terms. This is done as follows:

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��Along the left boundary and the lower boundary of the domain, a three point forward difference formula

is used to discretize the spatial derivatives of the velocity terms. This is done as follows:

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��Thus the values of the right hand side of the equation are computed at all the points.

The Press0 technique:

In this approach, the spatial derivatives are discretized using a backward difference formula at all the

interior points. However, the values of the right hand side of the equation are set to zero along all four

boundaries.

The Press0 technique is effective in maintaining the stability of the scheme at higher Reynolds numbers.

At an Re of 100, the Press0 technique fares quite well, whereas the Press technique is not stable, causing

the solution to go out of bounds if run for even a moderately high number of time steps. However, at

lower Re, the Press technique seems to give more accurate solutions, especially with respect to the flow

rate calculations. This is further discussed in subsequent sections of the report.

Once both the sides of the equation have been discretized, an LU decomposition method is used to solve

the system of linear equations. The resulting values of pressure are used in the second half of the pressure

step to calculate the values of u**

and v**

, using the previously mentioned expressions:

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The spatial derivatives of pressure are discretized using a central difference scheme as follows:

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��This results in the following algorithm:

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The values of u**

and v**

calculated in this step are passed to the viscous step.

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Two Viscous Steps:

The two viscous steps take into account, the viscous terms in the Navier-Stokes equation. When viewed in

isolation, this step is indeed the solution of a two dimensional diffusion equation. The viscous step is

further split into two steps so that each step only deals with one dimension. The solution of the first step is

passed as known constants to the equations of the second step, which upon being solved, yield the final

solution.

The First Viscous Step:

In the first viscous step, the following equations are solved:

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These equations are discretized using the Crank-Nicolson scheme i.e.

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The second order spatial derivative terms are discretized using a three point central difference scheme as

follows:

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Substituting this in the previous formula yields the expressions:

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Upon simplification, they yield the following algorithms:

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*��+�� '�� � ��������

These can be written mathematically as a set of linear equations. Illustratively, for the equations solving

the streamwise velocity components, we have the following expression:

,-./�0 � ���Where [A] is a matrix of coefficients of size [(nx-1)×(ny-2)], {u} is a matrix of unknowns of size [(nx-

1)×1], and {R} is a matrix of constants of size [(nx-1)×1].

The system of equations is solved using the Thomas algorithm. This algorithm is implemented using the

subroutines BANFAC and BANSOL.

The equations are solved at all the internal points and at the outflow boundary. At the outflow boundary,

the Neumann boundary conditions are applied for the streamwise velocity, as follows:

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Which gives, ����� � ����

Hence, these values are substituted at the outflow boundary.

While solving the first viscous step, the equations are solved first for a fixed value of y and all points

along x from x=2 to x=nx. The value of y is then incremented and again the equations are solved for all

the aforementioned values of x. In this manner, the entire grid is solved for, except for the inlet boundary,

and the channel walls where Dirichlet Boundary conditions apply.

The matrix equations for the equations that give v***

, are as follows:

,-./�0 � ���Where [A] is a matrix of coefficients of size [(nx-2)×(ny-2)], {v} is a matrix of unknowns of size [(nx-

2)×1], and {R} is a matrix of constants of size [(nx-2)×1].

As there are no Neumann boundary conditions applicable to the vertical component of velocity, these

equations are only solved at internal points and Dirichlet boundary conditions are applied at the

boundaries. These equations are also solved using the Thomas algorithm.

The intermediate values u***

and v***

that are obtained from the first viscous step act as the right hand side

of the equations that finally solves for the values of un+1

and vn+1

in the second viscous step.

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The second viscous step:

The second viscous step solves the equations,

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This is solved using a Crank-Nicolson method, as was the case in the first viscous step.

After applying the Crank-Nicolson scheme, the following expressions are obtained: ,k

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The second order spatial derivative terms are discretized using a three point central difference scheme as

follows:

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By applying this discretization and simplifying the expressions, we obtain the following equations:

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(� � '��)��� �� � '����� ��� � '����� ���� � (� � '��)��� ��� � '����� ���� � '����� �����

*��+�� '�� � ��������

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These can be written mathematically as a set of linear equations. Illustratively, for the equations solving

the streamwise velocity components, we have the following expression:

,-./�0 � ���Where [A] is a matrix of coefficients of size [(nx-2)×(ny-2)], {u} is a matrix of unknowns of size [(nx-

2)×1], and {R} is a matrix of constants of size [(nx-2)×1].

The system of equations is solved using the Thomas algorithm. This algorithm is implemented using the

subroutines BANFAC and BANSOL.

The equations in the second viscous step are first solved at a fixed value of x, and for all values of y from

y=2 to y=ny-1. Subsequently the value of x is incremented and the equations are once again solved for all

the aforementioned values of y.

The equations that solve for vn+1

, have a similar matrix representation and are solved in the same way.

The values, un+1

and vn+1

, represent the final values of the streamwise and vertical components of velocity

at the end of one complete time step. For the next time step, these values are passed back to the subroutine

that performs the convection step. The entire process is repeated until convergence is achieved.

Flow-Rate Calculation:

The flow rate is computed at each point along the channel length by numerically integrating along the y-

direction. The numerically calculated flow rate is then compared to the constant inlet flow rate and an rms

error (q-rms) is calculated.

The numerical integration is performed using the built-in Matlab function trapz .

trapz calculates the area under a curve by splitting it up in to the sum of the areas of trapeziums.

Notation:

u-rms is the RMS error calculated by comparing the numerically calculated velocity values at the outflow

with the exact solution.

q-rms is the RMS error calculated by comparing the numerically calculated flow rate at every point in the

x-direction with the constant flow rate at the inlet of the channel.

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Results and Plots:

Results are first presented for the standard case where in xmax=3, dx=dy=0.1, and dt=0.01. This case is

tested for two values of Reynolds number i.e. Re=1 and Re=10. The solution has been obtained using

both the Press and the Press0 technique. Results for both are presented, discussed and compared in the

following sections.

For Re=1:

Results for the Press0 Technique:

Re CFL ∆t

Time steps to

convergence Tolerance u-rms q-rms

1 0.1 0.01 100 1.00E-10 0.0119 0.015

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Results for the Press Technique:

Re CFL ∆t

Time steps to

convergence Tolerance u-rms q-rms

1 0.1 0.01 88 1.00E-10 0.0062 0.0025

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For Re=10:

Results for the Press0 Technique:

Re CFL ∆t

Time steps to

convergence Tolerance u-rms q-rms

10 0.1 0.01 600 1.00E-10 0.0039 0.0021

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Results for the Press Technique:

Re CFL ∆t

Time Steps to

Convergence Tolerance u-rms q-rms

10 0.1 0.01 696 1.00E-10 0.0057 0.0014

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Discussion of results and comparison between the Press and Press0 technique:

Initially, it was decided that only one of the methods would be implemented and presented in the

report. However, a comparison between the two methods seemed to provide an interesting

insight into the numerical scheme under examination.

As the results will indicate, the Press0 technique fared poorly at lower Reynolds number

whereas the Press technique did quite well at Re=1, producing a significantly lower rms error for

the streamwise velocity component. However at higher values of Re, the Press0 technique fared

better than the Press technique. However, at all the values of Re, the Press technique produced

better results for the flow rate and these were closer to the constant flow rate value at the inlet.

The major difference between the two solution methods was evident at the highest Reynolds

number that was tested for in this project. At Re=100, the Press technique failed completely and

caused the solution to go out of bounds, a clear sign of instability. The Press0 technique

however, produced good results even at this high Reynolds number.

The reason for this difference between the two schemes can be attributed to the fact that the

Press technique assigns definite values to the boundary nodes in the calculation of the

perturbation pressure matrix. These definite values seem to be getting larger and larger at the

high Reynolds number, resulting in the solution going out of bounds. In fact, it was observed that

even the Press0 technique produced the best results when run for a coarser mesh, and when the

channel length was extended significantly.

One interpretation of this result could be that, at this high Reynolds number, with the refinement

of the grid, the increase in round-off error was over-powering the decreased discretization error,

hence resulting in a solution that tended to go out of bounds. This case could be run on a faster,

more capable computer, where in it would be possible to run the program with a higher precision

and hence lower round-off errors, and only then can this theory about the instability of the code

be validated. Another more obvious reason for the solution going out of bounds would be the

explicit nature of the scheme applied in the convection step, which is only conditionally stable

and tends to become unstable at higher values of Re.

The best results obtained for Re=100 have been presented in subsequent sections. Also,

improved results at Re=1 and Re=10, owing to mesh refinement and extension of the domain

length in the x-direction have been provided.

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Results for Re=100 using the Press0 technique:

xmax Grid Time steps to convergence u-rms q-rms

60 21x21 35000 0.0046 0.0032

70 21x21 35000 0.0036 0.0024

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Best Results for Re=1:

Using the Press technique:

For xmax=3, dx=0.05, dy=0.05, Press technique

Re CFL ∆t Time steps to convergence Tolerance u-rms q-rms

1 0.1 0.01 157 1.00E-10 0.005 2.80E-03

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Best Results for Re=10:

For xmax=3, dx=0.05, dy=0.05, dt=0.01, Press technique

Re CFL ∆t Time steps to convergence Tolerance u-rms q-rms

10 0.1 0.01 1145 1.00E-10 0.0027 6.04E-04

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Results using the Press0 technique:

For xmax=3, dx=0.05, dy=0.05, dt=0.01, Press0 technique

Re CFL ∆t Time steps to convergence Tolerance u-rms q-rms

1 0.1 0.01 154 1.00E-10 0.0041 0.0042

10 0.1 0.01 1003 1.00E-10 0.0014 4.11E-04

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Results obtained for various channel lengths:

Press0 Technique

xmax dt Grid Re Number of iterations to convergence u-rms q-rms

5 0.01 51x21 10 578 0.0036 0.0018

5 0.01 101x41 10 965 0.0011

3.24E-

04

10 0.01 101x21 10 549 0.0035 0.0015

Press Technique

xmax dt Grid Re Number of iterations to convergence u-rms q-rms

5 0.01 51x21 10 671 0.0057 0.0014

5 0.01 101x41 10 1100 0.0027 6.28E-04

10 0.01 101x21 10 638 0.0058 0.0015

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Some additional plots:

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Vector Plot for a simple 11x11 grid

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Scope for further work:

As the overall method of solution chosen to solve this problem was quite simplistic, the scope for

further work is quite large. Only some of these avenues are touched upon in this report.

• The numerical scheme might be run on a more capable computer to reduce round off

errors. This would help the code to perform more satisfactorily at higher Reynolds

numbers and if there are any complications in the geometry of the system.

• The usage of more accurate schemes is another way in which improvement can be done.

The use of explicit schemes can also be substituted for schemes which are stable for a

wider range of parameters.

• The implementation of boundary conditions can be done more accurately, with a reduced

number of assumptions, hence producing a more realistic model of the flow.

• A finite volume method can be implemented to model the same flow.

• The use of a locally refined mesh can help to better capture the intricacies of the flow. To

be more precise, a more refined grid nearer to the channel walls will help to more

accurately model the boundary layer that is formed in that region.

There are many other ways in which to increase the efficacy of the numerical solution scheme

presented in this report. The author plans on pursuing these avenues in the near future.

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A listing of the code and a description of the various subroutines:

Name Type of Script Function

Main.m Main Program Setting of input parameters,

Calculation of matrices

required by the different sub-

routines, testing for

convergence and calculation

of RMS error. Uses the

subroutine Press.m for the

pressure step.

Main0.m Main Program Same function as Main.m. The

only difference being that it

calls Press0.m for the pressure

step.

Convec.m Subroutine Solves the convection step of

the solution as described in the

report.

Press.m Subroutine Solves the pressure step of the

solution using the Press

technique, as described in the

report.

Press0.m Subroutine Solves the pressure step of the

solution using the Press0

technique.

visc.m Subroutine Solves the two viscous steps

in the solution as described in

the report.

redif.m Subroutine Solves the right hand side of

the first viscous step.

redif2.m Subroutine Solves the right hand side of

the second viscous step.

flowrate.m Subroutine Calculates the flow-rate and

also calculates the rms error

by comparing the numerically

calculated flow rate with the

constant flow rate at the inlet.

Plotter.m Subroutine Plots all the necessary graphs.

fact.m Subroutine Performs LU decomposition.

solve.m Subroutine Solves the Perturbation

Pressure equation by back

substitution.

banfac.m Subroutine Performs the forward sweep in

the Thomas Algorithm

bansol.m Subroutine Backward sweep of the

Thomas Algorithm