TURBULENCE MODELLING AND IT’S IMPACT ON CFD PREDICTIONS FOR COOLING OF ELECTRONIC

COMPONENTS

Kulvir K Dhinsa, Chris J Bailey, Koulis A Pericleous Centre for Numerical Modelling and Process Analysis

University of Greenwich, Old Royal Naval College

Greenwich, London, SE10 9LS, UK Phone: +44(0)20-8331 8141 Fax: +44(0)20-8331 8665

Email: k.k.dhinsa@gre.ac.uk

ABSTRACT

This paper will discuss Computational Fluid Dynamics (CFD) results from an investigation into the accuracy of several turbulence models to predict air cooling for electronic packages and systems. Also new transitional turbulence models will be proposed with emphasis on hybrid techniques that use the ε−k model at an appropriate distance away from the wall and suitable models, with wall functions, near wall regions. A major proportion of heat emitted from electronic packages can be extracted by air cooling. This flow of air throughout an electronic system and the heat extracted is highly dependent on the nature of turbulence present in the flow. The use of CFD for such investigations is fast becoming a powerful and almost essential tool for the design, development and optimization of engineering applications. However turbulence models remain a key issue when tackling such flow phenomena. The reliability of CFD analysis depends heavily on the turbulence model employed together with the wall functions implemented. In order to resolve the abrupt fluctuations experienced by the turbulent energy and other parameters located at near wall regions and shear layers a particularly fine computational mesh is necessary which inevitably increases the computer storage and run-time requirements. The PHYSICA Finite Volume code was used for this investigation. With the exception of the ε−k and

ω−k models which are available as standard within PHYSICA, all other turbulence models mentioned were implemented via the source code by the authors. The LVEL, LVEL CAP, Wolfshtein, ε−k , ω−k , SST and klk /ε models are described and compared with experimental data. KEY WORDS: Thermal Management, Heat Transfer, Low Reynolds Number (Transitional) Flows, Flow Separation and Reattachment.

NOMENCLATURE a1 SST model constant, 0.31 C1ε k-ε model constant 1.44 C2ε k-ε model constant 1.92 Cµ k-ε model constant 0.09 CDkω cross diffusion term D maximum local length scale, m

E LVEL integrating constant, 9.0 G turbulent generation rate i internal energy k turbulent kinetic energy, m2s-2 k~ thermal conductivity, Wm-1K-1 L distance to the nearest wall, m lε turbulent dissipation length lµ turbulent mixing length p pressure, Pa Re Reynolds number S source term S SST model strain rate t time, s u velocity vector, ms-1 u, v, w velocity components, ms-1 u+ wall velocity y* Wolfshtein model Reynolds number y+ wall distance Greek symbols α revised LVEL model constant, 0.01 β k-ω model constant, 0.075 β* k-ω model closure coefficients β* SST model constant, 0.09 ε turbulent dissipation rate, m2s-3 κ von Karman constant, 0.41 Г diffusion coefficient µ dynamic viscosity, Nsm-2 µt-max maximum turbulent dynamic viscosity, Nsm-2 ν kinematic viscosity, m2s-1 ν+ effective viscosity ρ fluid density, kgm-3 σk k-ε model constant, 1.0 σk k-ω model constant, 2.0 σε k-ε model constant, 1.3 σω k-ω model constant, 2.0 Φ dissipation function φ general variable ω specific dissipation rate, s-1 Subscripts l laminar t turbulent + dimensionless

INTRODUCTION

The electronics industry is now developing at a rapid rate and with it the problems associated with the cooling of electronic packages is becoming increasing complex. The electronics packaging design community is now commonly using CFD to thermally characterize the performance of electronic packages and systems. These simulation tools solve both fluid flow and temperature throughout the system. For fluid flow the classical Navier Stokes equations are solved, with suitable turbulence models, to accurately capture the flow around a package. Flow characterization within electronic products is represented by the relationship between the turbulent phenomena and the flow structures within the wake of an electronic package. The association between the flow structure and the generation of heat encourages the flow to separate and recirculate locally between each board-mounted package. The consequence of these features has the potential to cause the electronic product to malfunction due to the temperature exceeding some operational or reliability limit. Extensive documentation has been published which characterizes the flow structures present when studying fluid flow around a surface mounted cubical obstacle, whether referring to a single or matrix array of obstacles. The interested reader is referred to the works of Meinders and Hanjalić [1], who concentrate on low Reynolds number flow. Also the work of Martinuzzi and Tropea [2] and Hussein and Martinuzzi [3] who consider higher Reynolds number flow phenomena. In duct flows, if the Reynolds number (Re) is below 1000 the flow is said to be laminar; hence the viscous forces dominate. Above Re = 4000 the inertia forces dominate over the viscous forces and flow perturbations grow, leading to turbulence. The range between the two limits is termed the transitional region. This is the region of most relevance to the electronics packaging community. The uncertainty of mathematically modelling turbulence is reflected in the large variety of models available. These range from Prandtl’s zero-equation mixing length model [4] to more complex higher order models. At present the vast majority of CFD calculations for this application area use Launder and Spalding’s ε−k model [5], which is optimized for high Reynolds number flows. For the purpose of this investigation CFD results will also be shown for the low Reynolds number implementation of the Wilcox ω−k model [6, 7], Menter’s SST model [8], which combines ε−k and ω−k , and a newly formulated two-layer hybrid klk /ε turbulence model for the prediction of flow around a heated package. It has now become a priority for electronic cooling CFD predictions to recognize the areas of low Reynolds number

flow within a system and to exploit transitional turbulence models which should ideally hold the attributes of being both accurate and computationally fast with regards to solution time [9]. Low Reynolds number turbulence modelling tends to be associated with the key disadvantage of requiring a fine computational mesh in near-wall regions, for example, this disadvantage holds true for the Wilcox ω−k turbulence model. This fine mesh constraint stems from the formulation of such models abandoning the use of wall functions which therefore require the solution of the viscosity-affected sub-layer close to the wall. Such techniques increase the computer storage and run-time requirements significantly greater than those methods which employ the wall function approach. In a typical congested environment of an electronics package, such attention to wall regions becomes impractical. Taking the above model disadvantage into consideration Eveloy et al. [10] recently published work comparing CFD predictions with experimental data for a particular set of components on a printed circuit board. These comparisons showed that the Shear-Stress Transport (SST) turbulence model [8], which is based on a hybrid approach formulation that employs the Wilcox ω−k model at near wall regions and the ε−k model at some distance away from the wall, results in better predictions for leading edge heat transfer and component thermal interaction. It was also demonstrated in [10] that the comparisons between CFD predictions and experimental data showed that the Spalart-Allmaras (SA) one-equation model [11] could be considered as a computationally cheaper alternative to the SST model. The aim of this research is to formulate a low Reynolds number turbulence model that can be used in a CFD code for accurate prediction of flow and temperature in electronic systems. To be applicable the model should:

• be relevant for congested domains containing a mix of low and high aspect ratio geometries as found in electronic systems (Figure 1)

• compute within a reasonable time, so that it can be used by electronic thermal design engineers.

Figure 1. Schematic of low/high aspect ratio geometries The formulated model will be compared with other models to see how it satisfies the above aim. This includes the k-ε, k-ω, SST and LVEL models although at present we are not investigating the SA model. This model requires the use of a ‘trip’ source in the eddy viscosity transport equation (see [11]) and hence the definition of a trip point in each boundary layer region. This may be practical for flat-plate type geometries but

lacks general applicability and should be used with caution for electronic systems where flow separation and re-attachment is important. Dhinsa et al. [12, 13] has demonstrated results for the Meinders single cube test configuration, using LVEL, LVEL-CAP, Wolfshtein, k-ε, and the k-ω models. In this paper details have also been presented on the development of the hybrid turbulence based on k-ε and k-l which at present looks encouraging. This paper extends this work by comparing the results for the Shear-Stress Transport (SST) model [14] against experimental data from Meinders [15].

MATHEMATICAL MODELS

This section will outline the formulation of all the turbulence models used in this investigation subject to the general conservation equation of the quantity φ . All simulations in this investigation have been constrained to steady state conditions therefore allowing the transient term to be ignored from equation (1), and all subsequent equations.

(1) With regards to equation (1), it has been necessary to hide uncommon terms for all the solved variables into the source term, a summary of which is given in Table 1.

Table 1. Governing equations of flow for a compressible Newtonian fluid

Equation φ Γφ Sφ

Continuity 1 0 0

x-Momentum

u

µ S

x

pMx+

∂

∂−

y-Momentum

v

µ S

y

pMy+

∂

∂−

z-Momentum

w

µ S

z

pMz+

∂

∂−

Energy

i k ~ Sup i+Φ+− div The source term contains any extra phenomena taking place in the system, such as the application of wall functions, gravitational and pressure effects. LVEL Turbulence Model This is a simple algebraic turbulence model which does not require the solution of any partial differential equations. The model depends on the calculation of the distance to the nearest wall, the local velocity and the laminar viscosity to determine the effective viscosity. The first step of the LVEL model is the solution of the following Poisson equation, which therefore allows the

calculation of the maximum local length scale and the local distance to the nearest wall to be completed with relative ease.

(2)

(3)

(4) With the local speed, V, and L a Reynolds number is calculated, then using the Reynolds number relationship stated below together with a version of the universal law of the wall the calculation of the dimensionless effective viscosity is obtain by analytical differentiation of the wall function relationship.

(5)

(6)

(7) This model results in a turbulent viscosity which varies from element to element. Further mathematical derivation of the model can be found in the work of Agonafer et al. [16]. Revised LVEL Model A single value for the turbulent viscosity based on a user specified velocity and length scale is calculated, but allows the value to vary according to the log-law of the wall for any elements close to solid surfaces. Model derivation is identical to the LVEL model discussed above, but in this instant an upper bound constrains the turbulent viscosity.

(8) Wolfshtein Turbulence Model The one-equation Wolfshtein turbulence model [17] solves a transport equation for the kinetic energy and then uses empirical functions to describe the turbulent mixing and dissipation lengths. A number of modifications have been suggested for the empirical functions used. One such model modification was formulated in 1975 by Norris and Reynolds [18], but will not be discussed any further in this investigation.

(9)

(10)

( ) ( ) ( )( ) Sut φφ φφρφρ

+=+∂

∂Γ graddivdiv

wall.at the 0 where1 =−=∇ φφ

φφ 22

+∇=D

φ∇−= DL

yu ++=Re

( ) 435.0ln1

min ,, =

= +++ κ

κyEyu

u

y

dd

v+

++ =

VLt ραµ =−max

( )

ρερ

σ

ρµρ

ρ

−

+=+∂

∂

+

Gv

kvkut

k

t

k

tl graddivdiv

lkC

ε

µε2343

=

(11)

(12)

Standard εk − Turbulence Model Launder and Spalding’s two-equation ε−k model is unarguably the most widely used and validated model employed for turbulent fluid dynamics to date. The extensive use of the model has highlighted both the capabilities and shortcomings of the model. The model has achieved notable success when dealing with thin shear layers and recirculating flows without the need for case-by-case modification of the model constants. Also success of the model is noted for confined flows where the normal Reynolds stresses are relatively unimportant compared to the Reynolds shear stresses which are of utmost importance. The model is favoured for industrial applications due to its relatively low computational expense and generally better numerical stability than more complex turbulence models such as the Differential Stress Equation Model (DSM) introduced by Launder et al. [19]. The predominant drawback of the standard ε−k turbulence model, for this application area, is that the model was designed for high Reynolds number flows therefore resulting poorly in terms of model accuracy when considering fluid flow over populated Printed Circuit Boards (PCB) which is usually classified as being low Reynolds number flow due to the small velocities and length scales encountered [20]. The formulation for Launder and Spalding’s turbulence model consists of two transport equations, one equation to describe the kinetic energy of turbulence and a second related to the rate of turbulent dissipation.

(13)

(14) With the eddy viscosity defined below.

(15)

Wilcox ωk − Turbulence Model This turbulence model was first introduced by Kolmogorov in 1942, a number of improvements have been suggested for the original model including those of Spalding [21] and Saiy [22]. The most popular suggestions for model improvements came from Wilcox [23] in 1988 who formulated a low Reynolds number alternative to the standard ε−k turbulence model. Wilcox suggested using an equation to represent the frequency of the vorticity fluctuations rather than using an equation describing the turbulent dissipation rate. The formulation of the turbulent kinematic viscosity and the closure coefficients can be found in [12, 23].

(16)

(17) It has been suggested by Menter [24, 25] that the core deficiency of the ω−k turbulence model stems from the sensitivity of the model to the free stream values of ω . A possible solution to this deficiency is to use a combination of the ω−k model equations implemented near wall regions and the ε−k turbulence model to be employed in the bulk flow region. This lead Menter to formulate the Shear-Stress Transport (SST) turbulence model [14, 26]. Shear-Stress Transport Turbulence Model The formulation of this two-equation hybrid type turbulence model is stated below.

(18)

(19) The blending function F1 is defined as:

(20)

lkCvt µµ2141=

( ) ( )** 016.0263.0 exp,exp 11 yy LL ll −− −− == κκ µε

( )

ρερ

σ

ρµρ

ρ

−

+=+∂

∂

+

Gv

kvkut

k

t

k

tl graddivdiv

( )

kC

kGvC

vut

t

tl

2

21

graddivdiv

ερ

ερ

εσ

ρµερ

ερ

εε

ε

−

+=+∂

∂

+

ερµ µ

2kCt =

( )

kGv

kvkut

k

t

k

tl

ωρβρ

σ

ρµρ

ρ

*

graddivdiv

−

+=+∂

∂

+

( )

ωβω

αρ

ωσ

ρµωρ

ωρ

ω

2

graddivdiv

−

+=+∂

∂

+

Gk

vut

tl

( ) ( ) ( )[ ]kkPkut

ktklk graddiv~div *

µσµωρβρρ

++−=+∂

∂

( ) ( )

( )[ ] ( ) ωω

σρωµσµ

βρωαρωρρω

ωω gradgrad1

12graddiv

div

21

22

⋅−+++

−=+∂

∂

kF

Sut

tl

=

4

22

2*14

,500

,maxmintanhLCD

k

L

v

L

kF

k

l

ω

ωσρ

ωωβ

(21) The turbulent eddy viscosity is calculated as follows:

(22) The second blending function F2 is defined by:

(23) To prevent the build-up of turbulence in the stagnation regions a production limiter is used

(24) The model constants are calculated using the F1 blending function

(25) Model constants stated below. The disadvantage of this model is highlighted when considering recirculating flow, separation and reattachment. Menter et al. [14]. Two-Layer Hybrid klkε Turbulence Model This newly formulated hybrid turbulence model exploits the advantages of the standard ε−k model by using the original model equations at a sufficient distance away from the enclosure walls and any electronic package and then drops down to a single equation turbulence model in the vicinity of solid devices. The model divides the test geometry into two regions; the allocation of the division is determined by using a critical Reynolds number. Region A solves the standard ε−k model equations; Region B on the other hand switches to the lk − model and solves an appropriate set of equations which represent the turbulent dissipation rate. The lk − model also employs the use of wall functions and damping functions to make this model economical in terms of run-time. To bridge the gap between the two turbulence models a matching technique has been used. The main advantage of such a hybrid model is that it is computationally cheap in terms of the mesh requirement

which therefore makes this a fast and economical model to run. Also the ε−k model advantages will filter through to the klk /ε model; these advantages include outstanding performance for many industrially significant flows such as confined flow phenomena. It should be noted that it has not yet been determined if the disadvantages of the standard ε−k model also filter through into the hybrid model. Initial model testing suggests that the disadvantages of the ε−k model will not impose themselves onto the hybrid model. Further model derivation can be found in the work of Dhinsa et al. [12].

MODEL CONFIGURATION Fluid flow and heat transfer CFD predictions around a single cube mounted on the base of a low Reynolds number enclosure are compared against Meinders’ experimental data. The numerical analysis has been undertaken using the finite volume CFD code PHYSICA [27], a multi-physics code developed at the University of Greenwich. The test channel has dimensions (1215 x 61 x 600)mm and uses an inlet velocity of 4.47m/s resulting in a low Reynolds number of approximately 4440 based on the cube height. The cube of size H = 15mm is mounted x/H = 50 downstream of the inlet boundary on the centreline allowing the use of a symmetry boundary condition in the z-direction. The test channel base plate is constructed from phenol-formaldehyde which is 10mm in thickness and has a thermal conductivity of 0.33W/mK. The structure of the cube can be decomposed into two materials; the core of the cube measuring 12mm is constructed from copper which is kept at a constant temperature of 75°C. The second material is an epoxy resin which encapsulates the copper and has a uniform thickness of 1.5mm. The base of the ‘cube’ does not have an epoxy layer attached therefore resulting in the overall dimensional structure of the cube measuring (15 x 13.5 x 15)mm, a schematic of which is shown in Figure 2.

Figure 2. Schematic representation of the test geometry

Copper Core 75°C

Epoxy Layer 1.5mm

x z

y

Flow

Symmetry boundary condition

Inflow

Outflow

856.0,5.00.1,85.0

,0828.0,403,44.0,95

2121

2121

====

====

σσσσ

ββαα

ωωkk

−⋅= 10,gradgrad

12max 10

2 ωω

ρ σ ωω kCDk

( )FSa

kavt

21

1

,max ω=

=

2

2*2500

,2

maxtanhωωβ L

v

L

kF

l

( )ωρβµ kPPx

U

x

U

x

UP kk

i

j

j

i

j

itk

*.10,min~ =→∂

∂+

∂

∂

∂

∂=

( )φφφ 2111 1 FF −+=

RESULTS AND DISCUSSION

The CFD predictions for this investigation have been conducted on a computational mesh density of approximately 350,000 elements. For any ω−k turbulence model simulations the computational mesh density has been increased to approximately 575,000 elements. This increased mesh density is necessary at locations of critical importance, such as around the cube, as the ω−k model does not utilize wall functions and therefore needs a relatively fine computational mesh at solid interfaces where viscous sub-layers exist. It should also be mentioned that a grading technique has been used to obtain a computationally fine mesh at any solid/fluid interface. The flow over surface mounted bluff bodies is often associated with the separation of shear layers at the side and top faces of the body in question. These shear layers may reattach either on the channel wall or between neighbouring obstacles when considering matrix arrays. It has been identified in previous work [1, 2, 3] conducted in this area that when simulations of this type of phenomena are being undertaken four main flow features are observed. With reference given to Figure 3, the most pronounced flow feature that emerges is the structure of the horseshoe vortex which originates upstream of the windward face and extends around both lateral sides of the cube whilst weakening in the streamwise direction. The flow in proximity to the horseshoe vortex is characterized as being in an unsteady turbulent state while the core flow in the corridor above the cube remains almost undisturbed. The separation of the top shear layer results in a bound recirculation vortex located at the leading top edge of the cube and is observed only for the first cube when considering an array or a matrix of cubes. The footprint left by the vortex is identified as being two counter rotating circles [28].

Figure 3. Schematic representation of the flow around a

surface mounted cube (Hussein & Martinuzzi [3]) The separated side shear layers are the result of high vorticity recirculation’s close to the leading edge. Their origins are

located at the channel floor with the vortex tubes covering a significantly large surface area of both the side faces. Also the vortex tubes are observed to be confined by the main flow and the presents of the cube. Finally an arc-shaped vortex confined to the depth and height of the cube dominates the wake flow. This vortex is caused by a strong up wash close to the leeward face. A similar counter rotating footprint, as that found on the top surface of the cube, is also observed. The post processing tool Tecplot [29] was used to identify the four recirculation regions with the aid of flow streamlines. The CFD simulated results are presented in Figure 4 for the two-layer hybrid klk /ε turbulence model. Referring to Figure 4 the streamline plots A and B represent the xy-plane with plot B focusing on the bound recirculation vortex. This visualization plane is taken at the location of the symmetry boundary condition. Plots C and D correspond to the xz-plane with plot D focusing on the side recirculation vortex tube. This plane is located at y/H = 0.5. It can be concluded at this point that all the flow features observed in Meinders experimental work have been reproduced in this investigation.

A) Horseshoe, bound and wake vortices.

B) Bound vortex.

C) Side and wake vortices.

D) Side vortex.

Figure 4. Streamline plots identifying the flow

recirculation regions CFD x-direction velocity results have been extracted for the nearest element centre of each control volume extending along the normal direction. The results, shown in Figure 5, for all turbulence models considered in this investigation have been validated against Meinders’ experimental data for the streamwise velocity profile in the normal direction at a

Side vortex recirculation tube

Arc-shaped vortex

Horseshoe vortex

Bound recirculation

location of x/H = 6.7 downstream of the inlet at the centreline of the channel.

Figure 5. x-velocity profile at a location of x/H = 5.7 downstream of the cubes leeward face.

There are several conclusions which can be drawn from the results presented in Figure 5. It can clearly be identified that for all turbulence models considered the greatest discrepancies are located at the wake region of the cube and at the top test channel plate. Secondly both the LVEL type models together with Wolfshtein’s model seem to produce the most accurate results, especially in the two problematic regions identified above. The SST model is still recovering after the wake recirculation region; this poor flow recovery rate has also been identified by Menter et al. [30]. Finally it has become evident that the standard ε−k turbulence model significantly over predicts the profile in the bulk flow, which suggests that this model experiences thicker boundary layers compared to the other models investigated. Also the model is noted to under predict in the wake region of the cube most severely in comparison to all other turbulence models. Surface temperature profiles have also been taken along the vertical path around the cube. It should be noted that the temperature results for the Wolfshtein one-equation turbulence model have not yet been completed and will therefore not be mentioned any further.

Figure 6. Surface temperature measurements along the vertical

path ABCD

CFD temperature results are located at the first element centre in the epoxy resin material, not on the surface itself which is where the experimental results are taken. As analysis is undertaken of the temperature profile shown in Figure 6 it is worth noting that if inaccuracies exist in the flow domain these will also filter through to the temperature domain, suggesting that an initial temperature result forecast can already be made for the standard ε−k model. It has been shown that the flow results for the ε−k model poorly predict the actual phenomena taking place within the system, this leads onto the proposal that the ε−k model is likely to demonstrate poor temperature predictions as with flow†. Considering the rear face (AB) of the cube in Figure 6 it can be concluded that the two-layer hybrid klk /ε turbulence model produces the best comparison with the experimental data, suggesting that the wake flow vortex that is resident in this region is accurately predicted. For the top face (BC) all turbulence models investigated fail to capture the sharp peak in the data which coincides with the bound recirculation region. Nevertheless both LVEL type models, ω−k and the SST turbulence model do seem to show a slight increase in temperature close to the top leading edge of the cube. Finally at the front face (CD) of the cube both LVEL type models and the two-layer hybrid klk /ε model predict the closest agreement at the mid face of the cube. It is evident that the standard ε−k model on the other hand performs poorly, noticeably at the top and front faces. It should be mentioned that the experimental data was collected with a technique referred to as Infrared Thermography (IR) which may have attributed to possible inaccuracies at the corners of the cube, due to the restriction of such a data retrieval system.

CONCLUSIONS The major contributor to electronic malfunction is the presence of the flow recirculation’s discussed in this investigation. As the recirculation vortices develop they encapsulate heat from a package and restrict further heat being extracted and removed from the system. From the results presented in this paper it can be concluded that the standard high Reynolds number ε−k turbulence model struggles to accurately predict real world phenomena even for simplistic cases. Other models investigated give better results, but no clear winner at present emerges.

† Although velocity data around the cube itself where not available at

the time of writing

This is work in progress, and further studies will concentrate on developing the two-layer hybrid klk /ε turbulence model which shows at this stage, considerable promise.

ACKNOWLEDGMENT This research, undertaken at the University of Greenwich, is financially supported by the Engineering and Physical Sciences Research Council (EPSRC) through the PRIME Faraday organization as an Industrial Case Award. The authors would also like to acknowledge helpful discussions with Flomerics Ltd. relating to this application field.

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