Conjugate Heat Transfer Finite Difference Method for Turbulent

MATH6640 – Numerical Methods for PDEs :
Final Project Report
Submitted by:
Nathan Rolander
Prepared for:
Dr. Yingjie Liu
Georgia Institute of Technology
Monday, December 06, 2004

Nathan Rolander MATH6640 – Final Project
Fall 2004
CONTENTS
NOMENCLATURE 4
INTRODUCTION 5
PROBLEM OVERVIEW 5
MOTIVATION 5
PROBLEM DESCRIPTION & GEOMETRY 6
MODEL DEVELOPMENT 7
FLUID FLOW 7
CA_POD_SVD 7
CA_EXTRACT 7
HEAT TRANSFER 8
CONTROL VOLUME BASED FINITE DIFFERENCE APPROACH 8
STAGGERED FLOW GRID 10
DISCRETIZATION APPROACHES 10
UPWINDING SCHEMES APPROXIMATION & STABILITY 11
UPWINDING SCHEMES ACCURACY 13
COMPUTING THE CELL PECLET NUMBER 14
SOLUTION METHODS 16
OVERALL ROUTINE 16
COLD AISLE SIMULATION 16
ADVECTION 17
CONSTRUCTING THE STIFFNESS MATRIX 17
BUILDK 18
FA 18
BOUNDARY CONDITIONS 18
APPLYBCS 19
SOLVE SYSTEM 19
LINESOLVER 19
TRANSFORMTEMP 20
THOMAS 20
SIMULATION ACCURACY 21
CONDUCTION ACCURACY 21
CONVECTION ACCURACY 23
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RESULTS 25
TEST 1 26
TEST 2 27
TEST 3 28
DISCUSSION 29
ALGORITHM SPEED 29
TRANSIENT STABILITY 29
SUMMARY 29
REFERENCES 30
APPENDIX 31
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Nomenclature
RANS - Reynolds Averaged Navier Stokes
k-, - k epsilon turbulence model,
k - turbulent kinetic energy,
, - turbulent dissipation
CFD - computational fluid dynamics
HVAC - Heating Ventilation and Air Conditioning
POD - Proper Orthogonal Decomposition
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Nathan Rolander MATH6640 – Final Project
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INTRODUCTION
In this project a two-dimensional convection diffusion thermal system corresponding to a
simulation of a data center cold aisle is presented. This results in a two-dimensional
steady state elliptic system which is solved using a finite difference approach. This
system has a degree of nonlinearity, requiring iteration of the system to achieve
convergence to steady parameter values.
PROBLEM OVERVIEW
A Data Center is a dedicated room of computers systems. These computers are racks in a
cabinet enclosure, and include servers, computational workstations, or switches for
communications. The heat load of these rooms can be 200kW, hundreds of times that of a
human occupied room. The computers require a cool, dry environment for best
performance, and therefore require a dedicated environmental conditioning system. This
system often uses as much as 40% of the power to the room just for HVAC. As the cost
reaches millions of dollars per year, efficiency is of utmost concern.
MOTIVATION
In order to improve cooling efficiency of the HVAC system, numerical thermal fluid
simulations are performed to determine superior cabinet configurations. The cabinet
geometry and air flow velocities of the data center create turbulent convective conditions,
leading to long convergence times using CFD solvers such as the k-, RANS turbulence
model. This high computational expense is compounded when a large number of design
variables are incorporated into the problem. The long computational time incurred makes
these tools, poorly suited for use in iterative algorithms employed in optimization
methods. In the preliminary stages of design, model fidelity can be traded-off for
computational speed in order to effectively investigate and map the feasible design space.
Therefore the development and application of computationally efficient reduced order
models is essential for effective design of complex thermal systems.
An emerging reduced order model development approach for turbulent convective flow is
proper orthogonal decomposition (POD). This method has been used successfully to
create low dimensional steady state flow models within a prescribed range of parameters.
The POD construct is used to build a two-dimensional convective flow simulation of aircooled
server cabinet located within a data center. This two dimensional model is
representative of a cross-section of a fully loaded cabinet. This cabinet simulation
contains four individual racks, each with a prescribed airflow rate and power generation.
With the development of this computationally efficient fluid flow model, a heat transfer
model is required to solve for the cabinet temperature profile. The temperature profile
dictates the operating temperature of the computer chips, and this whether they will
overheat. This thermal model development is presented in this project.
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Nathan Rolander MATH6640 – Final Project
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PROBLEM DESCRIPTION & GEOMETRY
The problem geometry, fluid flow, and thermal system to be simulated is shown below in
Figure 1. This geometry shows two server cabinets with four racks each on both sides.
The central flow path is called the cold aisle, and acts as the plenum to distribute the air
to the individual server racks. The flow through this aisle comes primarily from the
bottom, with continuity dictating the flow direction through the upper inlet yielding either
an inflow or outflow boundary condition. The fluid flow in the system is assumed to be
forced convection only, and thus independent from the thermal system. This allows the
energy equation can be solved independently for a prescribed flow field. The fluid flow
in the system is normally solved for using a linear combination of basis functions, that are
generated by Proper Orthogonal Decomposition of observations that are generated in
FLUENT, a commercial CFD program. However, in this project the FLUENT generated
flow data is used directly for simplicity as only the thermal system is being investigated.
System
Boundary
Vout 1
Vout 2
Vout 3
Vout 4
q 1, Tc 1
q 2, Tc 2
q 3, Tc 3
q 4, Tc 4
Left Server
Bank
Ambient Air
Too, Continuity
Cold Aisle
Vin, Tin
q 5, Tc 5
q 6, Tc 6
q 7, Tc 7
q 8, Tc 8
Right Server
Bank
Vout 5
Vout 6
Vout 7
Vout 8
System
Boundary
Figure 1 -Problem geometry and physics
The thermal model to be developed to obtain the temperature profile of the system
depicted in Figure 1 above must solve for both convection and diffusion over a 2D
domain. Because the air that leaves though the arrows labeled Vout is recycled to some
degree thought the top inlet, the double arrows labeled Continuity, Too is dependent upon
the temperature leaving the domain. In this manner, these two variables are coupled
creating a nonlinear problem to solve for the steady state Too condition. This recirculation
of this exhausted air is to varied from 0% to 100% to represent the efficiency
of the hot air removal system of the data center.
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Nathan Rolander MATH6640 – Final Project
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MODEL DEVELOPMENT
FLUID FLOW
In order to determine the sole accuracy of the thermal model being developed the flow
fields to be used are the direct output of the FLUENT generated data sets. These
simulations are performed using the k-, RANS turbulence model with a range of
boundary conditions creating a window of flow observations that represent what would
typically occur within a data center such as the one being modeled. These flow data sets
are read into MATLAB and processed for use by the thermal model by the function
ca_pod_svd and ca_extract, the headers and descriptions are shown below. The full
programs with detailed comments are available at the end of this report in the Appendix.
ca_pod_svd
function [x2,y2,u,v,um,vm] = ca_pod_svd
%CA_POD.M Extract FLUENT Observations & perform POD
% This function extracts node based FLUENT data representing
% the flow field of teh cold aisle geometry. In this project
% this data is used directly, however in my research the
% Proper Orthogonal Decomposition of the observations is
% applied to generate basis functions called POD modes
% of the flow data for use in my reduced order flow model.
% This program is based off of work by Jeff Rambo.
%
% Outputs:
% x2 - vector of FLUENT x-coordinate grid data
% y2 - vector of FLUENT y-coordinate grid data
% u - vector of FLUENT u-flow field data
% v - vector of FLUENT v-flow field data
% um - vector of POD mode u-flow field data
% vm - vector of POD mode v-flow field data
ca_extract
function [x2,y2,u2,v2,r] = ca_extract(name)
%CA_EXTRACT Extract and Process FLUENT data files
% Function to extract data for the cold aisle model reduction
% study. Call [x,y,u,v,dimrot]=ca_extract(name) where
% 'name' is the Fluent output ASCII file name. 'x','y','u','v'
% are the nodal coordinates and velocities, 'P' and 'nu' are
% the pressure and effective viscosity, respectively. 'dimrot'
% is vector index numer to pivot about in the vec2mat command
% for reshaping vectors into matrices for field plotting.
% This program is based off of work by Jeff Rambo.
%
% Inputs:
% name - name of the FLUENT ASCII file name
%
% Outputs:
% x2 - vector of FLUENT x-coordinate grid data
% y2 - vector of FLUENT y-coordinate grid data
% u2 - vector of FLUENT u-flow field data
% v2 - vector of FLUENT v-flow field data
% r - length of y vector for creating matrix data
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HEAT TRANSFER
The two dimensional conductive and convective heat transfer problem forms an elliptic
partial differential equation, commonly reffered to as as the energy equation, given below
in equation 1.
d d ⎛ dφ ⎞ d ⎛ dφ
⎞
( ρφ) + ⎜ρuφ−Γ ⎟+
⎜ρvφ−Γ
⎟ = S
(1)
dt dx ⎝ dx ⎠ dy ⎝ dy ⎠
In this equation S is the volumetric heat generation, φ is the temperature, ρ is the fluid
density, u and v are the flow velocities in the x and y directions respectively, Γ is the
thermal conductivity k. In this problem the transient term goes to 0 as it is a steady state
problem. It is also convineint to lump the conductive and convective flux terms together.
This makes the calculation of the numerical flux balance easier and is important for
stability reasons to be explored later. This results in the following equation form given
below in equation 2.
dJ dJ
dx dy
x y
+ = S
(2a)
dφ
Jx= ρuφ −Γ (2b)
dx
dφ
Jy= ρφ v −Γ (2c)
dy
Control Volume based Finite Difference Approach
The basis of the numerical method is the conversion of the general differential equation 1
to an algebraic equation relating the temperature of the point under consideration, P, to
the temperatures of the surrounding points N, E, S, W. This results in a flux balance into
and out of the control volume as shown in Figure 2 below.
Figure 2 - Example control volume
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These flux equations are given in equation 2. However, there are multiple ways to
discretize them. There are explored in the next section, however, before this it is
important to define the dimensionless numbers that are used in these discretization
schemes and help to characterize the system.
Diffusion Conductance
The diffusion conductance D is the quantity of heat flux from the diffusive term of the
equation. The subscript De is the diffusion conductance across the control volume face e,
as shown in Figure 2. The diffusion conductance De is given by equation 3a. This is a
more accurate approximation to the average thermal conduictivity between nodes P and E
over a simple average. However, in this project the thermal condictivity of the air is
taken to be constant because of the low temperature variation and thus equation 3a
simplifies to equation 3b.
Mass Flow Rate
D = A
D
e e
e
1
( δx ) ( δx
)
e
+
Γ Γ
e e− e e+
P E
(3a)
Γ
= (3b)
δ x
The mass flow rate is F is the quantity of mass flowing across a boundary. The subscript
Fe is the mass flow rate across the control volume face e and. The mass flow rate Fe is
obtained from equation 4 shown below.
Peclet Number
Fe = ( ρu)
eAe (4)
The Peclet number is defined locally across each face of the control volume, yielding a
maximum cell Peclet number. The general form of the Peclet number is shown in
equation 5a and the Peclet number specific to the face e, Pee, in equation 5b.
uL
P ρ
= (5a)
Γ
Pe
F
e
e = (5b)
De
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Nathan Rolander MATH6640 – Final Project
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Equation 5 shows that the Peclet number is in fact the ratio of conductive flux to diffusive
flux. This is a critical parameter in the analysis of the possible discritization schemes
explored next.
Staggered Flow Grid
The u variable, the velocity across the face of the element is read in from the generated
velocity field by the flow model. However, because there needs to be a velocity at all
four sides of each element where the temperature is computed, a staggered flow grid is
employed. This is shown below in Figure 3. Here the grey area is the element of
constant temperature, the node at the center is the point where temperature is computed.
The intersecting dashed lines are where the u and v components of the velocity are
computed by the flow model. This results in the temperature matrix being one element
smaller in each dimension, as well as the “artifact” of the grid that is the missing u, v
point at the top right hand corner.
Discretization Approaches
Figure 3 - Staggered grid diagram
To discretize the equation is space, the coefficients to multiply the surrounding nodes by
must be found. This general discretization is given in equation 6a, where a a represents
the coefficients of the surrounding nodes and b is equal to the heat flux input to the
element.
a φ = a φ + a φ + a φ + a φ + b
(6a)
P P E E W W N N S S
The following equations give the general form of the coefficients for each side of the
control volume. These coefficents are functions of the dimensionless parameters defined
ab , represents
earlier, and a function of the Peclet number A(|Pe|) . In these equations
the maximum of a and b.
( ) ,0
a = D A Pe + − F
(6b)
E e e e
( ) ,0
a = D A Pe + F
(6c)
W w w w
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( ) ,0
a = D A Pe + − F
(6d)
N n n n
( ) ,0
a = D A Pe + F
(6e)
S s s s
aP= aE + aW + aN + aS
These coefficeints a are related to the fluxes into the element Jx and Jy given in equation 2
through equation 7, shown in general form as the flux through the control volume face
between two adjacent elements. Here the capitalized letters represent opposite face
coefficients from adjacent nodes.
*
J B i A i 1
= φ − φ +
(7)
Substituting one of equations 6b-f with an appropriate function A(|Pe|) and equation 7
yields the function for the total flux entering a specific control volume face. There are
many approaches to approximating the function A(|Pe|), these are all dependent upon the
Peclet number, Pe. This choice of approximation also determines the accuracy and the
stability of the method, as investigated next.
Upwinding Schemes Approximation & Stability
In order to determine the most accurate and stable approximation of the flux function J
though the there are several schemes for the function A(|Pe|). These are based upon the
Taylor series expansion of equation 2, an approximation or exact solution to the
differential equation between two nodes, or a combination of these. These schemes are
presented below in equations 8.
Scheme Formula for A(|Pe|)
The value of these functions A(|Pe|) versus the input Peclet Number is shown below in
Figure 4.
(6f)
Central difference 1− 0.5 Pe
(8a)
Upwind 1 (8b)
Power law ( ) 5
0, 1− 0.1 Pe
(8c)
Exponential (exact)
Pe
( )
exp Pe − 1
(8d)
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Nathan Rolander MATH6640 – Final Project
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A(|Pe|)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Central Difference
Flux Function A(|Pe|) for Various Schemes
Central difference
Upwind
Power law
Exponential (exact)
0
0 1 2 3 4
|Pe|
5 6 7 8
Figure 4 - Flux function A(|Pe|) versus Peclet number
The central difference scheme is based upon a piece-wise linear profile for φ between
nodes. Viewing the plot above all schemes produce a physically realistic solution, except
for the central difference scheme which produces values outside of the [0,1] range when
the Peclet number is greater than 2. This means that the mesh would have to be fine
enough to keep the grid Pe < 2 so keep the scheme stable, as Pe is defined on a local
scale.
Upwind
The upwind scheme, or upwind-difference scheme, assumes that the value of φ at the
interface is equal to φ at the upwind node. This assumption creates a more stable scheme,
however its accuracy is questionable, as to be shown next.
Exponential
The exponential scheme is based upon the analytical solution to equation 2b, which is the
equivalent to substituting equation 8c into equation 6. This scheme therefore gives the
exact solution regardless of the grid spacing, however the exponential function is
computationally expensive to compute.
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Power law
The power law scheme is an approximation to the exponential scheme that produces very
close results. An added advantage beyond the increased computational speed is that if a
fluid flow speed of 0 is encountered the equation does not fail, as the exponential scheme
does.
Upwinding Schemes Accuracy
With the stability of the schemes established, their accuracies must be established. This
is done through the application of a simple one dimensional system given below in
equation 9.
φ = φ + φ (9a)
aPP aE E aW W
aE
φ P =
a + a
E W
In this system the diffusion D is taken to equal 1 thus φ is a function of Pe only. φW is
taken as 0 while φE is taken to be 1. This results in the following values of φ for a wide
range of Pe shown below in Figure 5.
φ p
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
Prediction of φ p by Various Schemes
Central difference
Upwind
Power law
Exponential (exact)
-10 -8 -6 -4 -2 0
Pe
2 4 6 8 10
(9b)
Figure 5 - Prediction of φP by various schemes
This plot shows that the power law is an almost exact approximation to the exact solution
generated by the exponential scheme. This plot also shows that the upwind scheme
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Nathan Rolander MATH6640 – Final Project
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yields good results when Pe < 1, and the upwind scheme when Pe > 10. Therefore the
selection of scheme depends upon the cell Peclet number as given by the fluid properties
and the FLUENT generated flow field.
Computing the Cell Peclet Number
For this simulation and all further runs, the following fluid parameters are used for air at
300K, given by Incopera and DeWitt [3]. Because the bulk temperature of the fluid does
not change very much, these constant properties are an accurate approximation.
Property Value
Density, ρ 1.1614 kg/m 3
Thermal conductivity, k 26.3e-3 W/m 2
Of the 48 FLUENT generated flow fields, the more extreme inlet and outlet velocity
cases were selected that would yield the higher cell Peclet numbers. The flow field
selected is shown below in Figure 6.
Figure 6 - Flow field for computation of cell Peclet number
This yields a cell Peclet number distribution as shown below in Figure 7.
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Y Axis (m)
2
1.5
1
0.5
Cell Peclet Number Distribution
0.5 1 1.5 2 2.5 3
X Ax is (m)
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Figure 7 - Cell Peclet number distribution
Viewing the figure above, the maximum cell Peclet number occurs in the regions of
highest flow velocity, as to be expected. The highest numbers reach values of 4.5.
Viewing Figure 5 shows that this is the region of maximum error for the upwind and
central difference schemes. Furthermore, reducing the mesh size is undesirable because
of the increased computational time as well as the need to interpolate the fluid flow field
introducing erroneous data!
Therefore the A(|Pe|) function selected is the power law, as given in equation 8c.
Substitution of this equation into equation 7 yields the final flux across the control
volume interface e given by equation 10 below.
{ ( ) }
JA= Fφ + DA Pe + [ −F ,0] ( φ − φ )
(10)
e e e P e e e P E
Substitution of this same A(|Pe|) function into equation 6 yields the final discrimination,
used in the finite difference schemes presented next.
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SOLUTION METHODS
Three different solution methods were used in this project and compared with regard to
their speed. All methods are implicit, the solutions obtained through the inversion of the
resulting NxM by NxM stiffness matrix. The methods to be used are direct inversion of
the stiffness matrix through Gauss Elimination and the alternating direction line by line
iterative method. The direct inversion method is to be applied to the full matrix as well
as a “sparse” matrix, a bandwidth reduced matrix format used by MATLAB for
comutational efficincy.
OVERALL ROUTINE
The overall routine of constructing and solving the matrix system is performed by the
function Advection. This function solves for the thermal profile of the data center cold
aisle for a perscirbed upper air inlet temperature. This function is itterated by the
function ColdAisleSimulation. The headers and descriptions of both functions are given
below. The complete code for all functions in this report are at the end of this report in
the Appendix.
Cold Aisle Simulation
function [T,X,Y,x,y,u,v,u2,v2,x2,y2]
= ...
ColdAisleSimulation(Q,eta,method,plots,x2,y2,u2,v2)
%COLDAISLESIMULATION Convection Simulation of Cold Aisle
% This program computes the temperature field of the
% cold aisle of a data center. The bulk of the code is
% in the function Advection, which is iterated here
% to find the appropriate inlet temperature.
%
% Inputs:
% Q - power generated from each PCB server board
% eta - ratio of heat energy recycled into system
% method - solution algorithm to be implemented
% 'full': direct inversion of full matrix
% 'sprs': direct inversion of sparse matrix
% 'line': alternating line by line TDMA algorithm
% plots - number of plots to be generated
% 1: Temperature profile
% 2: Vector flow field
% 3: Cell Peclet number
% 4: Streamlines & velocity partidcles
% x2 - vector of FLUENT x-coordinate grid data
% y2 - vector of FLUENT y-coordinate grid data
% u2 - vector of FLUENT u-flow field data
% v2 - vector of FLUENT v-flow field data
%
% Outputs:
% T - matrix of computed nodal temperature data
% X - vector of ascending unique x coordinates
% Y - vector of ascending unique y coordinates
% x - matrix of sorted x-coordinate grid data
% y - matrix of sorted y-coordinate grid data
% u - matrix of sorted u-flow field data
% v - matrix of sorted v-flow field data
% x2 - vector of FLUENT x-coordinate grid data
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Nathan Rolander MATH6640 – Final Project
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% y2 - vector of FLUENT y-coordinate grid data
% u2 - vector of FLUENT u-flow field data
% v2 - vector of FLUENT v-flow field data
Advection
function [T,X,Y,x,y,u,v,dT2] = ...
Advection(x2,y2,u2,v2,Q,dT,eta,method,plots)
%ADVECTION Numerical Energy Equation Solver with Advection
% This function computes the temperature field of the cold
% aisle of a data center using a given flow field. The
% function can use one of three solution methods, direct
% matrix inversion, sparse matrix inversion, and alternating
% line by line solution using the TDMA algorithm. The
% resulting temperature, flow, Peclet number, and streamline
% data can be plotted. The energy continuity is computed
% allowing for the iteration of the function to converge
% to a steady state solution.
%
% Inputs:
% x2 - vector of FLUENT x-coordinate grid data
% y2 - vector of FLUENT y-coordinate grid data
% u2 - vector of FLUENT u-flow field data
% v2 - vector of FLUENT v-flow field data
% Q - power generated from each PCB server board
% dT - top inlet air temperature rise
% eta - ratio of heat energy recycled into system
% method - solution algorithm to be implemented
% 'full': direct inversion of full matrix
% 'sprs': direct inversion of sparse matrix
% 'line': alternating line by line TDMA algorithm
% plots - number of plots to be generated
% 1: Temperature profile
% 2: Vector flow field
% 3: Cell Peclet number
% 4: Streamlines & velocity partidcles
%
% Outputs:
% T - matrix of computed nodal temperature data
% X - vector of ascending unique x coordinates
% Y - vector of ascending unique y coordinates
% x - matrix of sorted x-coordinate grid data
% y - matrix of sorted y-coordinate grid data
% u - matrix of sorted u-flow field data
% v - matrix of sorted v-flow field data
% dT - top inlet air temperature rise from continuity
CONSTRUCTING THE STIFFNESS MATRIX
The stiffness matrix is constructed by a routine that scans through each line of the flow
field matrix and fills in the computed coefficents from equation 6 into the appropriate
elements. This function BuildK and its subfunctions fa header and description is given
below.
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BuildK
fa
function [K,B,Peclet] = BuildK(N,M,u,v,dx,dy,Tin,method)
%BUILDK Constructs Stiffness Matrix for Heat Transfer Problem
% Scans through the flow field of the given geometry and
% applies the correct cefficeints into the stiffness matrix
% to be solved implicitly either by direct matrix inversion
% or a line by line technique. This function apples insulation
% boundary conditions to all geometry edges.
%
% Inputs:
% N - number or elements in x
% M - number of elements in y
% u - vector of u-flow field data
% v - vector of v-flow field data
% dx - delta x
% dy - delta y
% Tin - top inlet air temperature
% method - solution algorithm to be implemented
% 'full': direct inversion of full matrix
% 'sprs': direct inversion of sparse matrix
% 'line': alternating line by line TDMA algorithm
%
% Outputs:
% K - stiffness matrix
% B - heat flux vector
% Peclet - matrix of cell Peclet numbers
function [fa,PE] = fa(u,L,A,s)
%FA Determines stiffness matrix coefficeint
% This function computes the discrete flux coefficients
% into the control volume (CV) under consideration. This is
% computed using the power law upwinding approximation,
% using the Peclet number, Convective flux, and Diffusion
% flux into the element. These are computed using
% seperate subfunctions.
%
% Inputs:
% u - scalar flow magnitude into control volume
% L - width of CV w.r.t. face under consideraton
% A - height of CV w.r.t. face under consideraton
% s - sign to be used in Power Law approximation
%
% Outputs:
% fa - flux coefficient
% PE - cell Peclet number
BOUNDARY CONDITIONS
The boundary conditions were applied as adiabatic, outflow, or with an input flux either
from the air inlet at a specified temperature, or from the heat generated from the surface
of the racks. These were implemented through changing the appropriate coefficient of
the flux between elements. There are two functions to perform this operation, ApplyBCs
and ApplyBCs_rot, which applies the same boundary conditions to the system that has
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been rotated 90 degrees for use by the alternating direction line by line solution method.
The function header and description of ApplyBCs is given below.
ApplyBCs
function [K,B] = ApplyBCs(K,B,N,M,u,v,dx,dy,Tin,q,Delta_T)
%APPLYBCS Applies Boundary Conditions to Equation System
% This function modifies the system stiffness matrix to
% apply the specific boundary conditions of the cold aisle
% simulation. This includes the adiabatic PCB boards and
% associated heat flux inputs as well as the inflow and
% outflow boundaries.
%
% Inputs:
% K - stiffness matrix
% B - heat flux vector
% N - number or elements in x
% M - number of elements in y
% u - vector of u-flow field data
% v - vector of v-flow field data
% dx - delta x
% dy - delta y
% Tin - top inlet air temperature
% Q - power generated from each PCB server board
% dT - top inlet air temperature rise
%
% Outputs:
% K - corrected stiffness matrix
% B - corrected heat flux vector
SOLVE SYSTEM
The Gauss Elimination of the full stiffness matrix is performed using the internal
MATLAB routines for both the full and sparse matrix systems. The alternating line by
line solution works by looking at a single line and setting the flux input to elements from
the lines above and below the line to constants using the most recently computed
temperature values. This creates a tridiagonal matrix system which is solved for by the
Thomas algorithm, the header of which is given below as Thomas. This is then repeated
for orthogonal lines, in this case alternating between lines of x and y. In order to
alternate lines, two stiffness matrices are constructed, one for the normal geometry and
second for the geometry rotated 90 degrees. This allows the same solution function to be
applied, as shown in Advection. A transformation algorithm is then applied to transform
the solution temperature vector into the order required for multiplication with the rotated
stiffness matrix for the next iteration of the routine. The header of this function
TransformTemp is given below, as is the function LineSolver which is the entire routine.
LineSolver
function T = LineSolver(M,N,K,b,temp)
%LINESOLVER Line by Line Numerical Solution Method
% This function solves the banded matrix system that
% is formed from two dimensional implicit schemes.
% It solves each line seperately using the TDMA
% algorithm using the latest values of T to compute
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Nathan Rolander MATH6640 – Final Project
Fall 2004
% the flux inputs from above and below the line.
% This function can be used with alternating x and y
% direction for generally increased convergence
% speed.
%
% Inputs:
% N - Number of elements in x
% M - Number of elements in y
% K - Stiffness Matrix
% b - Heat flux input vector
% temp - temperature input vector
%
% Outputs:
% T - temperature output vector
TransformTemp
function tr = TransformTemp(M,N)
%TRANSFORMTEMP Transform Temperature Vector for Next Iteration
% Computes the vector of indices of the computed Temperature
% vector such that these indeces return the Temperature
% vector such that it can be multipled by the rotated
% stiffness matrix. This allows the itteration between the
% x and y line by line solution method. Can be use to compute
% transformation and reverse transformation.
%
% Inputs:
% N - number or elements in x
% M - number of elements in y
%
% Outputs:
% tr - vector of indices for transformation of T
Thomas
function x = Thomas(A,b,N)
%THOMAS Thomas Algorithm for Solving Tridiagonal Systems
% This function is the Thomas algorithm for solving tridiagonal
% systems of the form Ax = b. In this project it is used to
% solve for the temperature distribution along a single line
% of the computaitional domain.
%
% Inputs:
% A - tridiagonal matrix
% b - right hand vector
% N - length of matrix side
%
% Outputs:
% x - solution
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Nathan Rolander MATH6640 – Final Project
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SIMULATION ACCURACY
In order to determine if my simulation is accurate I have compared it to an analytically
computed conduction problem and a commercial CFD program called FEMLAB’s
solution to a convective problem. This is because there are no analytical solutions to a
problem that is similar enough to mine to test my simulation against.
CONDUCTION ACCURACY
To test the simulations accuracy of conduction only the output is compared to an
analytical solution to a square region with one side at higher temperature at steady state
as follows in equation 11.
Txx + Tyy
= 0
(11)
The analytical series solution to this problem is given below in equation 12, where λnL =
nπ.
∞ 1−cos( λnL)
T( x, y) = T0 + 2( TL −T0) ∑ sinh( λnx)sinh( λny)
(12)
λ Lsinh( λ L)
n= 1 n n
The resulting temperature distribution is plotted in Figure 8 below.
y-position (m)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Temperature Distribution
0
0 0.2 0.4 0.6 0.8 1
x-position (m)
Figure 8 – Conductive test temperature distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
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Nathan Rolander MATH6640 – Final Project
Fall 2004
The comparison of the numerical simulation with the analytical solution for lines of y and
x are shown in Figure 9 and Figure 10 respectively below.
Temperature (C)
Temperature (C)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Comparison of Numerical Conduction Model
y = 0.1
y = 0.25
y = 0.5
y = 0.1
y = 0.25
y = 0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x-position (x)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 9 - Comparison model for constant y
Comparison of Numerical Conduction Model
x = 0.25
x = 0.5
x = 0.75
x = 0.25
x = 0.5
x = 0.75
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y-position (m)
Figure 10 - Comparison model for constant x
Viewing the figures above it is shown that the results match perfectly, showing the
simulation represents the conductive flux excellently.
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Nathan Rolander MATH6640 – Final Project
Fall 2004
CONVECTION ACCURACY
The convective case selected was made similar to the system being simulated. Thus a
square region measuring 2m by 2m with a vertical flow of 0.5m/s with an inlet
temperature of 25 degrees C was subjected to a heat flux of 0.2W/m on the right and
0.4W/m on the left. These conditions were set up identically in both FEMLAB and the
finite difference simulation. The resulting temperature profile is shown below in Figure
11
Y Axis (m)
40
35
30
25
20
15
10
5
Temperature Field
5 10 15 20
X Ax is (m)
25 30 35 40
160
140
120
100
Figure 11 - Convective test temperature profile
The comparison of the numerical simulation with the analytical solution for lines of y and
x are shown in Figure 12 and Figure 13 respectively below.
80
60
40
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Nathan Rolander MATH6640 – Final Project
Fall 2004
Temperature (C)
Temperature (C)
110
100
90
80
70
60
50
40
30
Comparison of Numerical Convection Model
y = -0.5
y = -0.5
y = 0
y = 0
y = 0.5
y = 0.5
20
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x-position (m)
44
42
40
38
36
34
32
30
28
26
Figure 12 - Comparison model for constant y
Comparison of Numerical Convection Model
x = -0.5
x = -0.5
x = 0
x = 0
x = 0.5
x = 0.5
24
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
y-position (m)
Figure 13 - Comparison model for constant x
The figures above show an almost perfect match, except for some slight deviation along
the boundary. This is to be expected from the comparison of different numerical schemes
employing different solution techniques, however this is still a very strong result that the
simulation matches a commercial CFD package.
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Nathan Rolander MATH6640 – Final Project
Fall 2004
RESULTS
The complete simulation only has to iterate three times to converge to a steady solution to
overcome the nonlienarity incurred by the variable upper air inlet temperature. This
convergence is shown below in Figure 14.
Inlet Temperature 9C)
10
9
8
7
6
5
4
3
2
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Itteration Number
Figure 14 - Convergence of the Advection simulation routine
The simulation was then run for three different cases representing a low flow velocity
case, a high side velocity case, and a strong mixed velocity case. These cases would
yield very different temperature profiles and thoroughly test the convergence speed
capabilities of the different methods. The nomenclature used here is “Full” for the full
direct matrix inversion method, “Sparse” for the sparse matrix inversion method, and
“Line” for the line by line method.
The convergence time is measured until the change in delta t < 0.01 C between iterations
of Advection. The timing was performed with the internal MATLAB function tic toc
with multiple runs to ensure accurate results.
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Nathan Rolander MATH6640 – Final Project
Fall 2004
TEST 1
Convergence in 3 iterations.
Full: Convergence took 7.375 seconds.
Sparse: Convergence took 0.046 seconds.
Line: Convergence took 129 iterations in 1.680 seconds.
Y Axis (m)
2
1.5
1
0.5
Temperature Field
0.5 1 1.5 2 2.5 3
X Ax is (m)
30 35 40 45 50 55 60 65 70
Figure 15 - Temperature field
Figure 16 - Flow field
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Nathan Rolander MATH6640 – Final Project
Fall 2004
TEST 2
Convergence in 3 iterations.
Full: Convergence took 7.390 seconds.
Sparse: Convergence took 0.047 seconds.
Line: Convergence took 67 iterations in 0.938 seconds.
Y Axis (m)
2
1.5
1
0.5
Temperature Field
0.5 1 1.5 2 2.5 3
X Ax is (m)
26 28 30 32 34 36 38 40 42 44 46
Figure 17 - Temperature field
Figure 18 - Flow field
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Nathan Rolander MATH6640 – Final Project
Fall 2004
TEST 3
Convergence in 3 iterations.
Full: Convergence took 7.375 seconds.
Sparse: Convergence took 0.047 seconds.
Line: Convergence took 143 iterations in 1.890 seconds.
Y Axis (m)
2
1.5
1
0.5
Temperature Field
0.5 1 1.5 2 2.5 3
X Ax is (m)
30 35 40 45 50 55 60 65
Figure 19 - Temperature field
Figure 20 - Flow field
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Nathan Rolander MATH6640 – Final Project
Fall 2004
DISCUSSION
ALGORITHM SPEED
Analysis of the results presented in the previous section show that the Line-by-Line
solution is considerably faster than the direct matrix inversion. However, when the
sparse matrix bandwidth reduction function is employed, the inversion becomes much
faster because of the vast memory savings. However, this has been a useful learning and
coding experience for me.
TRANSIENT STABILITY
In this project the transient part of the energy equation has been ignored because of the
interest in the steady state solution. However, expanding the system to a transient
analysis is straightforward. The discretization of the time derivative can be done using an
explicit, implicit, or Crank-Nicholson scheme. Howver, for stability reasons the explicit
scheme must obey the CFL condition for it to be stable, given buy the courant number
shown in equation 13 below.
∆t
Cr = u (13)
∆ x
This equation is also true of the y-direction, with v and ∆y substituted for the x terms.
SUMMARY
Overall, I feel that the numerical scheme presented in this report is an excellent energy
equation solver for convective flow fields. This will aid me significantly with my future
research work on simulation based design of data center server cabinets.
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Nathan Rolander MATH6640 – Final Project
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REFERENCES
1. S. V. Patankar and D. B. Spalding, “A Calculation Procedure for Heat, Mass and
momentum Transfer in Three-Dimensional Parabolic Flows,” Int. J. Heat Mass
Transfer, 15, 1787-1806.
2. Patankar, S. V. Chapter 6 - Elliptic Systems: Finite-Difference Method I
Department of Mechanical Engineering, University of Minnesota – Twin Cities,
Minneapolis, Minnesota 55455
3. Rambo, Jeffrey “Model Reduction of Multiscale Turbulent Convection:
Application to Data Center Thermal Management”, PhD proposal, G. W.
Woodruff School of Mechanical Engineering, Georgia Institute of Technology
4. Incropera, F., Dewitt, D., “Fundamentals of Heat and Mass Transfer,” 2002
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Nathan Rolander MATH6640 – Final Project
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APPENDIX
The compilation of all of my MATLAB functions are compiled here in the order in which
they are called.
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