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Saha, Suvash, Brown, Richard, & Gu, YuanTong(2012)Prandtl number scaling of the unsteady natural convection boundary layeradjacent to a vertical flat plate for Pr>1 subject to ramp surface heat flux.International Journal of Heat and Mass Transfer, 55(23 - 24), pp. 7046-7055.

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https://doi.org/10.1016/j.ijheatmasstransfer.2012.07.017

https://eprints.qut.edu.au/view/person/Saha,_Suvash.htmlhttps://eprints.qut.edu.au/view/person/Brown,_Richard.htmlhttps://eprints.qut.edu.au/view/person/Gu,_YuanTong.htmlhttps://eprints.qut.edu.au/51494/https://doi.org/10.1016/j.ijheatmasstransfer.2012.07.017

Prandtl number scaling of the unsteady natural

convection boundary layer adjacent to a vertical flat

plate for Pr > 1 subject to ramp surface heat flux

Suvash C. Saha1 Richard J. Brown Y. T. Gu

School of Chemistry, Physics & Mechanical EngineeringQueensland University of Technology

GPO Box 2434, Brisbane QLD 4001, Australia

Abstract

It is found in the literature that the existing scaling results for the bound-ary layer thickness, velocity and steady state time for the natural convec-tion flow over an evenly heated plate provide a very poor prediction of thePrandtl number dependency of the flow. However, those scalings provide agood prediction of two other governing parameters’ dependency, the Rayleighnumber and the aspect ratio. Therefore, an improved scaling analysis usinga triple-layer integral approach and direct numerical simulations have beenperformed for the natural convection boundary layer along a semi-infiniteflat plate with uniform surface heat flux. This heat flux is a ramp function oftime, where the temperature gradient on the surface increases with time upto some specific time and then remains constant. The growth of the bound-ary layer strongly depends on the ramp time. If the ramp time is sufficientlylong, the boundary layer reaches a quasi-steady mode before the growth ofthe temperature gradient is completed. In this mode, the thermal bound-ary layer at first grows in thickness and then contracts with increasing time.However, if the ramp time is sufficiently short, the boundary layer developsdifferently, but after the wall temperature gradient growth is completed, theboundary layer develops as though the startup had been instantaneous.

Keywords: Boundary layer, vertical plate, ramp heat flux, Prandtl number.

1Corresponding author: Suvash C. Saha, Email: s c saha@yahoo.com

Preprint submitted to International Journal of Heat and Mass Transfer July 6, 2012

mitaTypewritten TextAccepted for publication in the International Journal of Heat and Mass Transfer

Nomenclature

g Acceleration due to gravityH Length of the plateA Dimensionless width of the domainB Dimensionless height of the domainP Dimensional pressurep Dimensionless pressurePr Prandtl numberRa Rayleigh numbert Dimensional timetw Dimensional ramp timets Dimensional steady state timeT Dimensional temperature of the fluidTw Dimensional temperature scale on the plateTws Dimensional temperature scale at quasi-steady stageTwq Dimensional temperature scale at quasi-steady modeu, v Dimensionless fluid velocities in the x- and y- direction respectivelyU, V Dimensional fluid velocities in the X- and Y - direction respectivelyVm Dimensional maximum velocityVms Dimensional velocity scale at quasi-steady stageVmq Dimensional velocity scale at quasi-steady modeVmw Dimensional velocity scale at steady state stagevm Dimensionless maximum velocityvms Dimensionless velocity scale at quasi-steady stagevmq Dimensionless velocity scale at quasi-steady modevmw Dimensionless velocity scale at steady state stagex, y Dimensionless Cartesian coordinatesX, Y Dimensional Cartesian coordinates

Greek lettersβ Thermal expansion coefficient∆T Temperature differenceδinn Dimensional viscous inner layer thicknessδinns Dimensional quasi-steady viscous inner layer thicknessδinnq Dimensional viscous inner layer thickness at quasi-steady modeδinnw Dimensional steady viscous inner layer thickness∆inn Dimensionless viscous inner layer thickness

2

∆inns Dimensionless quasi-steady viscous inner layer thickness∆innq Dimensionless viscous inner layer thickness at quasi-steady mode∆innw Dimensionless steady viscous inner layer thicknessδT Dimensional thermal layer thicknessδTs Dimensional quasi-steady thermal layer thicknessδTq Dimensional thermal layer thickness at quasi-steady modeδTw Dimensional steady thermal layer thickness∆T Dimensionless thermal layer thickness∆Ts Dimensionless quasi-steady thermal layer thickness∆Tq Dimensionless thermal layer thickness at quasi-steady mode∆Tw Dimensionless steady thermal layer thicknessδv Dimensional viscous layer thicknessδvs Dimensional quasi-steady state viscous layer thickness∆v Dimensionless viscous layer thickness∆vs Dimensionless quasi-steady state viscous layer thicknessΓw Heat fluxκ Thermal diffusivityρ Density of the fluidν Kinematic viscosityθ Dimensionless temperatureθw Dimensionless temperature scale on the plateθws Dimensionless temperature scale at quasi-steady stageθwq Dimensionless temperature scale at quasi-steady modeτ Dimensionless timeτw Dimensionless ramp timeτs Dimensionless quasi-steady time

1. Introduction

Natural convection and heat transfer of the boundary layer along a ver-tical flat plate is a classical problem (Jaluria, 1980; Gebhart et al., 1988;Hyun, 1994; Bejan, 1995). It is a common phenomenon in nature which isrelevant to industrial systems such as heat exchangers, electronic cooling,crystal growth procedures, etc. Several methodologies have been used toobserve the boundary layer development along the vertical plate. Recently,scaling analysis is widely used to predict the flow behavior and heat transferof different stages of transient flow development. The results of scale analysis

3

play an important role in guiding both further experimental and numericalinvestigations. It is a cost-effective way that can be applied for understandingthe physical mechanism of the fluid flow and heat transfer.

Patterson and Imberger (1980) conducted a pioneering scaling analysison the transient behavior of the flow of a differentially heated cavity. Theauthors classified the flow development through several transient flow regimesinto one of three steady state types of flow based on the relative values ofthe Rayleigh number Ra, the Prandtl number Pr, and the aspect ratio A.Later, scaling analysis was performed for various thermal forcing conditions,e.g. sudden temperature variations (Saha et al., 2011, 2010a,b,c; Lin et al.,2009; Bednarz et al., 2009, 2010), surface heating/cooling due to radiation(Lei and Patterson, 2002, 2005), uniform surface heat flux (Armfield et al.,2007; Lin and Armfield, 2005; Lin et al, 2008; Aberra et al., 2008; Saha etal., 2012) etc. Scaling is also used for different geometries, e.g. rectangularand triangular cavities, vertical and inclined flat plates, cylindrical enclosure,etc.

It was found in the literature that most of the scaling studies were con-ducted for an instantaneous thermal condition of either the isothermal orisoflux boundary condition on the wall which is not physically achievable.Therefore, there is a need to consider the case where the heating changeswith time initially. Saha et al. (2007) were the first who introduced ramptemperature boundary conditions to perform scaling analysis for the bound-ary layer adjacent to an inclined flat plate. Later, Patterson et al. (2009) andSaha et al. (2010a,b,c) followed the same approach for different geometries.However, the study for the ramped heat flux condition, where the temper-ature gradient changes initially with time and remains constant when theramp is finished, is still unrevealed. Therefore, it is the aim of this study toperform scaling analysis of the boundary layer adjacent to a vertical platedue to this boundary condition.

For the ramp heating temperature condition on both the vertical andinclined plates, two scenarios can be observed: (a) the flow enters into thequasi-steady mode before the ramp is finished and (b) the flow becomessteady state after the ramp is finished. The scaling results for the latter caseare exactly the same as for the instantaneous heating case. In the formercase, the boundary layer reaches a quasi-steady state before the temperaturegrowth is completed. In this mode the thermal boundary layer at first growsin thickness and then contracts with time and the fluid acceleration alsochanges character. However, when the ramp is finished, the flow becomes

4

steady state completely. The steady state values of the scaling results areexactly the same as in the case of instantaneous heating. The most importantpart of this boundary layer growth is between the quasi-steady state and thetime when the ramp is finished.

It was found in the previous scaling that the existing scaling relations ofthe thickness, velocity and transitional time of the thermal boundary layeradjacent to an evenly heated flat plate do not provide a good prediction ofthe Prandtl number dependency of the flow. Recently, a modification of theexisting scaling is performed where a triple layer integral method was used(Saha, 2011a; Saha et al., 2011; Saha, 2011b). The new, improved scalingcan now handle the Pr dependency very well. Especially, the viscous layerthickness scale, which is measured from the plate to the place where thevelocity is maximum, can be treated very well for different Prandtl numbers.However, the outer region of the viscous boundary layer still needs moreattention.

Most of the above scaling work was done in the context of an instan-taneous heating or cooling, that is, a step function, application of eitherthe isothermal or isoflux boundary condition on the vertical or inclined sur-faces. In reality, this is not possible to achieve physically (e.g. heat transferinto and out of a building). Therefore, it is necessary to consider the casewhere the heating or cooling are applied over some time period. Based onthis realization, various scaling laws are developed for the transient behav-ior of the unsteady natural convection boundary layer flow of an initiallyquiescent homogeneous Newtonian fluid with Pr > 1 along a vertical plateheated with a uniform ramp heat flux. The complex thermal forcing effect ishandled carefully and major flow scales are identified by using a triple-layerintegral approach. Furthermore, a series of Direct Numerical Simulationswith selected values of Ra and Pr in the ranges of 5× 107 ≤ Ra ≤ 109 and5 ≤ Pr ≤ 100 is carried out to verify various scaling laws obtained from thescaling analysis.

2. Problem formulation

Under consideration is the unsteady natural convection boundary-layerflow of an initially quiescent Newtonian fluid (with Pr > 1) adjacent to asemi-infinite vertical plate heated with a uniform ramp heat flux (see Fig.1). The fluid flow is also assumed to be two-dimensional. The temperature

5

gradient across the plate is Γw which increases linearly with time up to twand then remains constant.

The governing equations of motion are the Navier-Stokes equations ex-pressed in two-dimensional incompressible form with the Boussinesq approx-imation for buoyancy, which together with the temperature transport equa-tion are as follows,

∂U

∂X+

∂V

∂Y= 0 (1)

∂U

∂t+ U

∂U

∂X+ V

∂U

∂Y= −1

ρ

∂P

∂X+ ν

(∂2U

∂X2+

∂2U

∂Y 2

)(2)

∂V

∂t+ U

∂V

∂X+ V

∂V

∂Y= −1

ρ

∂P

∂Y+ ν

(∂2V

∂X2+

∂2V

∂Y 2

)+ gβ(T − T0) (3)

∂T

∂t+ U

∂T

∂X+ V

∂T

∂Y= κ

(∂2T

∂X2+

∂2T

∂Y 2

)(4)

Initially, the fluid is quiescent and isothermal at temperature T0. Theinitial conditions for velocity and temperature are then

U = V = 0, T = T0 ∀ X,Y, t < 0 (5)

On a semi-infinite vertical wall, the velocity boundary conditions are

U = V = 0, for X = 0, Y ≥ 0 (6)

The wall temperature gradient increases linearly from its initial value0 to the final value Γw at time tw, which is maintained thereafter. Thetemperature far from the plate is considered at T0.

It is well known that in natural convection the flow is governed by twonon-dimensional parameters, the Rayleigh number, Ra and the Prandtl num-ber Pr, where

Ra =gβΓwH

4

κν, and Pr =

ν

κ(7)

3. Scaling Analysis

When the ramp heat flux condition is applied on the plate, the tem-perature on the plate increases linearly which triggers the transient natural

6

convection phenomenon. A thermal boundary layer is developed adjacent tothe plate. To show the effect of the Prandtl number accurately it is nec-essary to examine the structure of the boundary layer in more detail. It isnoted that for higher Rayleigh number (Ra > 109), we may observe travellingwaves in the boundary layer [Xu-2009]. However, the Rayleigh number wechoose in this study is lower than that. The parameters characterizing theboundary layer development are predominantly the thermal boundary-layerthickness δT , the maximum vertical velocity um within the boundary layer,surface temperature Tw, the time ts for the boundary layer to reach steadystate, etc. For better understanding the flow field, a snapshot of isothermsand streamlines are plotted in Fig. 2 for Ra = 108 and Pr = 10 at differenttimes.

3.1. Early stage

The basic procedures described in Saha et al. (2012) are followed here butare appropriately modified for the case of a non-instantaneous temperaturegradient. The energy equation (4) indicates that since the fluid is initiallyquiescent, the heating effect of the plate will first diffuse into the fluid layerthrough pure conduction, resulting in a thermal boundary layer of thick-ness δT . Within the boundary layer, the dominant balance is between theunsteady and diffusion terms in the energy equation (4), that is,

δT ∼ κ1/2t1/2 (8)

This scaling is valid until the convection term becomes important. At thesame time the correct balance in the y−momentum equation (3) is betweenthe viscosity and the buoyancy (Saha et al., 2012).

0 ∼ ν ∂2V

∂X2+ gβ∆T

(t

tw

)(9)

where ∆T , the total temperature variation over the boundary layer, is of theorder O(ΓwδT ). Using (8) this may be written as

∆T ∼ Γwκ1/2t1/2 (10)

A typical temperature and velocity profiles adjacent to the semi-infiniteflat plate is shown in Fig. 3. Since the plate is regid and non-slip, the velocityof the fluid is zero on the plate surface. However, it increases from zero onthe vertical plate and reaches its maximum, which occurs within δT . The

7

velocity then decreases as the position is further from the plate. Note thatfor the case of Pr < 1 the scenario is different, which is outside the scopeof this study. Note that outside the thermal layer, the balance betweenviscosity and buoyancy is invalid. Instead, the fluid is driven by the diffusionof momentum by the viscosity from the region accelerated by buoyancy. Theviscous layer thickness is defined by the length scale δv. Therefore, we maydivide the whole boundary layer into three regions as shown in Fig. 3.

In regions I and II, the balance is between viscosity and buoyancy. How-ever, in region III the balance is between viscosity and inertia. In region I,the balance (9) gives

Vm ∼gβΓwδT

ν

(t

tw

)(δT − δi)2 (11)

In region II, the limit of the integral is taken between (δT − δi) and δT .

0 ∼ ν ∂V∂X

∣∣∣δTδT−δi

+ gβ

∫ δTδT−δi

TdX. (12)

Note that ∂V/∂X|δT−δi = 0 since the velocity is maximum there. Addi-tionally, we have

∂V

∂X

∣∣∣δT

∼ Vmδv − δT + δi

(13)

and ∫ δTδT−δi

TdX ∼ ∆T(

t

tw

)δi ∼ ΓwδT

(t

tw

)δi (14)

Hence,

Vm ∼gβΓwδT

ν

(t

tw

)δi (δv − δT + δi) (15)

Matching this with equation (11) obtained above for Vm gives

δi ∼δ2T

δT − δv(16)

As the buoyancy force is negligible in region III, the flow is driven solely bythe diffusion of momentum in which the unsteady term balances the viscousterm, yielding

8

Vmt

∼ ν Vmδ2v

(17)

further,

δv ∼ ν1/2t1/2 ∼ Pr1/2δT (18)

Hence, (16) becomes

δi ∼κ1/2t1/2

1 + Pr1/2(19)

Additionally, the length of the inner viscous layer (region I) is

δinn ∼ (δT − δi) ∼Pr1/2

1 + Pr1/2δT (20)

So the scaling (11) of Vm becomes

Vm ∼ Ra(

Pr1/2

1 + Pr1/2

)2 ( κH

)( ttw

)(t

H2/κ

)3/2(21)

Equation (21) is the scaling for maximum velocity (Vm) at the start-upstage. The flow in the period in which the initial thermal balance is betweenconduction and unsteady temperature growth is then described by the lengthscales (8) and (18), and the velocity scale (21). The temperature is describedby the scale O(ΓwδT t/tw), so long as t < tw.

3.2. Quasi-steady state

As time increases, the more heat is convected away. The boundary layerapproaches a steady state until convection balances conduction at time ts,i.e.

Vm∆T (t/tw)

H∼ κ∆T (t/tw)

δ2T(22)

where Vm and δT are calculated at ts. The relation (22) leads to a time scalewhen the boundary layer reaches a steady state

ts ∼1

Ra2/7

(H2

κ

)(tw

H2/κ

)2/7(1 + Pr1/2

Pr1/2

)4/7(23)

9

The corresponding maximum velocity scale at the steady state time is

Vms ∼ Ra2/7( κH

)(H2/κtw

)2/7(Pr1/2

1 + Pr1/2

)4/7(24)

The steady state thickness scale of the thermal boundary layer is

δTs ∼H

Ra1/7

(tw

H2/κ

)1/7 (1 + Pr1/2

Pr1/2

)2/7(25)

The scaling of the steady state inner viscous boundary layer thickness is

δinns ∼H

Ra1/7

(tw

H2/κ

)1/7(Pr1/2

1 + Pr1/2

)5/7(26)

The scaling of the steady state viscous boundary layer thickness is

δvs ∼H

Ra1/7

(tw

H2/κ

)1/7Pr1/2

(1 + Pr1/2

Pr1/2

)2/7(27)

The steady state temperature on the wall is then obtained from the thermalboundary layer thickness and the temperature gradient at the wall as

Tws ∼ΓwH

Ra1/7

(tw

H2/κ

)1/7 (1 + Pr1/2

Pr1/2

)2/7(28)

3.3. Quasi-steady mode, ts < t < tw

If tw > ts the boundary layer will reach a quasi-steady state at ts beforethe ramp is finished, and for ts < t < tw, the boundary layer will continue todevelop, governed by a balance between convection and conduction. Thus,for ts < t < tw, the boundary layer flow is also convecting heat away, and theboundary layer growth will change character when the convection balancesconduction, that is at time ts when

Vm∆T (t/tw)

H∼ κ∆T (t/tw)

δ2T(29)

For t > ts, δT is no longer governed by (8). This gives

Vm ∼κH

δ2T(30)

10

The same balances between buoyancy and viscosity still apply in regions Iand II, so that equation (16) holds. Further, since the boundary layer isin a quasi-steady state, the balance in region III is between advection anddiffusion of momentum, so that

Vm ∼νH

δ2v(31)

and again equation (19) holdsUsing this result the velocity given by the balance in region I is

Vmq ∼gβΓwδ

3T

ν

(t

tw

)(Pr1/2

1 + Pr1/2

)2(32)

Using (30) and (32), the δTq scale at the quasi-steady mode may be ob-tained as

δTq ∼H

Ra1/5

(twt

)1/5 (1 + Pr1/2

Pr1/2

)2/5(33)

The maximum velocity scale inside the boundary layer is

Vmq ∼ Ra2/5( κH

)( ttw

)2/5(Pr1/2

1 + Pr1/2

)4/5(34)

Corresponding scales for the viscous boundary layer thickness δv and theposition of the velocity maximum δinn are readily obtained. It is seen fromequations (33) and (34) that, in this quasi-steady stage of the boundarylayer development, the velocity increases, but the boundary layer thicknessdecreases with time. At t ∼ tw, the boundary layer becomes completelysteady, with thickness δTw and velocity Vmw given respectively by

δTw ∼H

Ra1/5

(1 + Pr1/2

Pr1/2

)2/5(35)

and

Vmw ∼ Ra2/5( κH

)( Pr1/21 + Pr1/2

)4/5(36)

The above discussion can be summarized in the following way: if theboundary layer reaches to the quasi-steady mode before the ramp is finished,

11

then the development of the boundary layer follows equation (8) which accel-erates according to equation (21) until time ts; it then interestingly contractsbut accelerates further in a quasi-steady mode until tw, following equations(33) and (34). When the ramp is finished the flow becomes completely steadyand is described by equations (35) and (36). However, if the steady state timeis longer than the ramp time the boundary layer follows equations (8) and(21) until the end of the ramp. At tw, the flow and temperature fields arethe same as for an instantaneous start up at the corresponding time, andany further development beyond tw is identical to that for an instantaneousstart-up (see Saha et al., 2012).

4. Normalization of the governing equations and the scaling

To verify the various scales, numerical solutions of the full Navier-Stokesand energy equations are obtained for a range of Ra and Pr values. Forconvenience, the non- dimensionalized forms of the governing equations areadopted

∂u

∂x+

∂v

∂y= 0 (37)

∂u

∂τ+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+ Pr

(∂2u

∂x2+

∂2u

∂y2

)(38)

∂v

∂τ+ u

∂v

∂x+ v

∂v

∂y= −∂p

∂y+ Pr

(∂2v

∂x2+

∂2v

∂y2

)+RaPrθ (39)

∂θ

∂τ+ u

∂θ

∂x+ v

∂θ

∂y=

(∂2θ

∂x2+

∂2θ

∂y2

)(40)

where x, y, u, v, θ, p and τ are the normalized forms of X,Y, U, V, T, P andt respectively, which are made normalized by the following set of expressions:

x =X

H, y =

Y

H, u =

U

κ/H, v =

V

κ/H, τ =

t

H2/κ, p =

P

ρκ2/H2, θ =

T

ΓwH(41)

It is noted that the origin of the coordinate system is located at the leadingedge of the heated plate.

12

The equations are solved on a domain −0.25 ≤ y ≤ B, 0 ≤ x ≤ Awhere A and B are the non-dimensional width and non dimensional heightrespectively. Domain dependency tests were carried out to ensure that thefar field boundary conditions did not affect significantly the detailed resultspresented below. The following boundary conditions, in non-dimensionalform, are applied

u = v = 0, for x = 0, y ≥ 0,Γw =

ttw, for x = 0, y ≥ 0 and 0 ≤ t ≤ tw,

Γw = 1, for x = 0, y ≥ 0 and t > tw,u = v = ∂T

∂x= 0 for x = 0, −0.25 ≤ y < 0,

∂u∂x

= v = ∂T∂x

= 0 for x = A, −0.25 ≤ y ≤ B,∂2u∂y2

= ∂2v

∂y2= ∂

2T∂y2

= 0 for 0 ≤ x ≤ A, y = B,u = v = ∂T

∂y= 0 for 0 ≤ x ≤ A, y = −0.25.

(42)

All scalings obtained above can be normalized based on the transforma-tion (41). However, selected normalized scales are presented here for brevity.

For τ < τs∆T ∼ τ 1/2 (43)

∆inn ∼ (δT − δi) ∼Pr1/2

1 + Pr1/2∆T (44)

vm ∼ Ra(

Pr1/2

1 + Pr1/2

)2τ 5/2

τw(45)

At τ = τs

τs ∼1

Ra2/7τ 2/7w

(1 + Pr1/2

Pr1/2

)4/7(46)

vms ∼ Ra2/7(

1

τw

)2/7(Pr1/2

1 + Pr1/2

)4/7(47)

∆Ts ∼τ 1/7

Ra1/7

(1 + Pr1/2

Pr1/2

)2/7(48)

∆inns ∼τ 1/7

Ra1/7

(Pr1/2

1 + Pr1/2

)5/7(49)

∆vs ∼τ 1/7

Ra1/7Pr1/2

(1 + Pr1/2

Pr1/2

)2/7(50)

13

θws ∼τ1/7w

Ra1/7

(1 + Pr1/2

Pr1/2

)2/7(51)

For τs < τ < τw

∆Tq ∼1

Ra1/5

(τwτ

)1/5 (1 + Pr1/2Pr1/2

)2/5(52)

∆innq ∼Pr1/2

1 + pr1/2∆Tq (53)

vmq ∼ Ra2/5(

τ

τw

)2/5 (Pr1/2

1 + Pr1/2

)4/5(54)

θwq ∼1

Ra1/5

(τwτ

)1/5(1 + Pr1/2Pr1/2

)2/5(55)

For τ > τw

∆Tw ∼1

Ra1/5

(1 + Pr1/2

Pr1/2

)2/5(56)

∆innw ∼Pr1/2

1 + pr1/2∆Tw (57)

vmw ∼ Ra2/5(

Pr1/2

1 + Pr1/2

)4/5(58)

5. Numerical procedure

Equations (38)-(40) are solved along with the initial and boundary con-ditions using the simple scheme. The Finite Volume scheme is chosen todiscretize the governing equations, with the quick scheme (see Leonard andMokhtari, 1990) approximating the advection term. The diffusion terms arediscretized using central-differencing with second order accuracy. A secondorder implicit time-marching scheme is also used for the unsteady term. Thedetailed numerical procedure can be found in Saha et al. (2010a,b,c).

Strong flows are present in the vicinity of the plate for the natural con-vection of a semi-infinite vertical flat plate. Therefore, a non-uniform rect-angualr finer mesh near the plate with an expansion factor of maximum 10%away from the wall is considered. This gives a grid size of 250 × 150. The

14

Run Ra Pr1 108 52 108 103 108 204 108 505 108 1006 5× 107 107 5× 108 108 109 10

Table 1: Values of Ra and Pr for eight simulations run

maximum non-dimensional time step is chosen as 10−6. Grid and time stepdependency tests are undertaken, with results obtained with half the min-imum grid sizes and expansion rates given above, and half the time steps.The variation between the results is negligible.

6. Results and discussion

Table 1 shows all simulations of the study. Runs 1− 5 are used to showdependence on Pr and Runs 2, 6− 8 show dependence on Rayleigh number.

In the following section, the velocity and temperature profiles are cal-culated at y = 0.5, which is sufficiently far from the leading edge and thedownstream end of the domain to avoid any end effects. The time series ofthe maximum vertical velocity (vm) is also recorded on the same line, whichis used to verify the velocity scaling relation.

Scaling relation (44) predicts that during the start-up stage the innerviscous boundary layer thickness ∆inn is dependent on Pr only. This scalingis validated by the numerical results (see Fig. 4). The profiles of the non-dimensional vertical velocity at different times during the start-up stage aredirectly plotted in Fig. 4(a). Now the velocity v and the distance x arenormalized by the scaling relations (45) and (44) respectively, which are re-plotted in Fig. 4(b). It is seen from Fig. 4(b) that the two scaling relationsbring all scaled profiles within the inner viscous boundary layers into a singleline at the start-up stage, which implies that (45) and (44) are good predictorsof the unsteady velocity and inner viscous layer thickness scales respectively.

The velocity profiles at two different times for each case are shown in Fig.5 for different Prandtl numbers and Rayleigh numbers in τs < τ < τw. Figure

15

5(a) shows the computed vertical velocity profiles calculated along y = 0.5.In Fig. 5(b), the velocity is normalized by its quasi steady scaling value vmqand the distance is normalized by its quasi-steady viscous layer thickness scale∆innq. Clearly, the scaling relations for the quasi-steady velocity scale (54)and viscous layer thickness scale (53) agree well with the numerical resultssince all profiles almost overlap onto a single curve in the inner viscous layer(Fig. 5b).

For the time period τ > τw, the boundary layer becomes completelysteady state. Figure 6 shows the velocity profiles for an arbitrarily selectedtime of τ > τw. Fig. 6(a) shows the dimensionless computed velocity profilesof eight simulations for all the parameters considered here along the sameline as before (y = 0.5). Then, the velocity is scaled by its steady state scalevmw given by equation (58) and the corresponding distance is scaled by ∆innwgiven by equation (57) and plotted in Fig. 6(b). Once again, the steady statevelocity scale (58) and viscous layer thickness scale (57) are confirmed by thesimulation results, as all the velocity profile curves fall on a single curve upto the position where the velocity is maximum.

The temperature profiles are calculated at the same time and on thesame line (y = 0.5) when the velocity profiles are drawn (Fig. 6) for τ > τw.The computed dimensionless temperature profiles are depicted in Fig. 7afor eight simularions run for all Rayleigh numbers and Prandtl numbers.Now, the distance is scaled by ∆Tw given by (56) and plotted in Fig. 7(b).The temperature profiles fall on a single curve for the entire length, whichconfirms that the thermal layer thickness scale (56) at the steady state stageis verified.

The time series of maximum vertical velocity is presented in Fig. 8. Thethree stages of the flow development can now be identified from the time seriesdata: the early stage, the quasi-steady stage and the steady-state stage. InFig. 8(a), the time series of the dimensionless maximum vertical velocitiesvm from all eight simulations are plotted. The velocity is then scaled by thequasi-steady velocity scale vms given by (47) and plotted in Fig. 8(b) againstthe time scaled by quasi-steady time scale τs given by (46). The locationof the end of the first stage on this plot in each case coincides, confirmingthat the scalings (46) and (47) are correct. Figure 8(c) shows the time seriesof maximum vertical velocity where the x-axis is (τ/τw)

2/5 and the y-axisrepresents the rest of the terms of (54). All curves meet at the place wherethe ramp is finished and fall together during the steady state stage. Thisconfirms the scaling relation of (54).

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Figure 9 illustrates the numerical results of the average surface temper-ature of the heated vertical plate. The computed time series of the surfacetemperature have been plotted in Fig. 9(a) for different parameters (Ra andPr). It is clear that there are significant effects of those parameters on thesurface temperature. Figure 9(b) represents the series of the surface temper-ature where the x-axis is (τ/τw)

1/5 and the y-axis contains the rest of theterms of equations (55). It is seen that all curves meet at the quasi-steadymode and at the steady state stage. This confirms the scaling relation oftemperature at the quasi-steady mode (55).

7. Conclusions

Natural convection due to ramp surface heat flux on a semi-infinite verti-cal flat plate is examined by scaling analysis and verified by direct numericalsimulations for various parameters considered here. The verification of thescaling relations includes thermal and viscous boundary-layer developmentsas well as surface temperature and several time scales. The flow developmentadjacent to the plate for this boundary condition depends on the comparisonof the time at which the ramp temperature gradient is completed with thetime at which the boundary layer completes its growth. It is revealed thatif the ramp time is longer than the steady state time, the thermal boundarylayer reaches a quasi-steady mode in which the growth of the layer is governedby the thermal balance between convection and conduction. However, if theramp is completed before the thermal boundary layer becomes steady, thesubsequent growth is governed by the balance between buoyancy and inertia,same as in the case of instantaneous heating. Numerical results demonstratethat the scaling relations are able to accurately characterize the physical be-havior in each stage of the flow. The present scaling analysis incorporates adetailed balance in the momentum equation depending on the thickness ofthe boundary layer which improves scaling predictions, especially where thePr variation effect is taken into account. The scaling relations are formedbased on the established characteristic flow parameters of the maximum ve-locity in the boundary layer (vm), the time for the boundary layer to reachthe quasi-steady mode (τs) and the thermal (∆T ) and viscous (∆v) bound-ary layer thickness. Through comparisons of the scaling relations with thenumerical simulations, it is found that the scaling results agree well with thenumerical simulations. It is also seen from the verification that the scalingworks well for the Pr dependency of the thermal layer thickness and the inner

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viscous layer thickness. However, for the outer layer a further improvementof the scaling is needed.

References

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Armfield, S.W. Patterson, J.C. Lin, W. 2007. Scaling investigation of thenatural convection boundary layer on an evenly heated plate. InternationalJournal of Heat and Mass Transfer 50, 1592–1602.

Bednarz, T.P., Lin, W., Patterson, J.C., Lei, C., Armfield, S.W. 2009. Scalingfor unsteady thermo-magnetic convection boundary layer of paramagneticfluids of Pr > 1 in micro-gravity conditions. International Journal of Heatand Fluid Flow 30, 1157–1170.

Bednarz, T.P., Lin, W., Saha, S.C. 2010. Scaling of thermo-magnetic con-vection, In Proceedings of the 13th Asian Congress of Fluid Mechanics,Dhaka, Bangladesh, 798–801

Bejan, A. 1995. Convection Heat Transfer, second ed., John Wiley & Sons,New York.

Gebhart, B., Jaluria, Y. Mahajan, R.L., Sammakia, B. 1988. Buoyancy-Induced Flows and Transport, Hemisphere, New York.

Hyun, J.M. 1994. Unsteady bouyant convection in an enclosure. Adv. HeatTransfer 24, 277-320.

Jaluria, Y. 1980. Natural Convection Heat and Mass Transfer, Pergamon,Oxford.

Lei, C., Patterson, J.C. 2002. Unsteady natural convection in a triangularenclosure induced by absorption of radiation. Journal of Fluid Mechanics460, 181–2009.

Lei, C., Patterson, J.C. 2005. Unsteady natural convection in a triangularenclosure induced by surface cooling. International Journal of Heat andFluid Flow 26, 307–321.

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Leonard, B.P., Mokhtari, S. 1990. ULTRA-SHARP Nonoscillatory Convec-tion Schemes for High-Speed Steady Multidimensional Flow.NASA TM1-2568 (ICOMP-90-12), NASA Lewis Research Centre.

Lin, W., Armfield, S.W. 2005. Unsteady natural convection on an evenlyheated vertical plate for Prandtl number Pr < 1. Physical Review E 72,066309.

Lin, W., Armfield, S.W., Patterson, J.C. 2008. Unsteady natural convectionboundary-layer flow of a linearly-stratified fluid with Pr < 1 on an evenlyheated semi-infinite vertical plate. International Journal of Heat and MassTransfer 51, 327–343.

Lin, W. Armfield, S.W., Patterson, J.C., Lei, C. 2009. Prandtl number scalingof unsteady natural convection boundary layers of Pr ¿ 1 fluids underisothermal heating. Physical Review E 79, 066313.

Patterson, J.C., Imberger, J. 1980. Unsteady natural convection in a rectan-gular cavity. Journal of Fluid Mechanics 100, 65–86.

Patterson, J.C., Lei, C., Armfield, S.W., Lin, W. 2009. Scaling of unsteadynatural convection boundary layers with a non-instantaneous initiation.International Journal of Thermal Sciences 48, 1843–1852.

Saha, S.C. 2011a. Scaling of free convection heat transfer in a triangularcavity for Pr > 1. Energy and Buildings, 43, 2908-2917.

Saha, S.C. 2011b. Unsteady natural convection in a triangular enclosure un-der isothermal heating. Energy and Buildings 43, 695–703

Saha, S.C., Patterson, J.C., Lei, C. 2011. Scaling of natural convection ofan inclined flat plate: Sudden cooling condition. ASME Journal of HeatTransfer 133, 041503.

Saha, S.C., Patterson, J.C., Lei, C. 2010a. Natural convection boundary layeradjacent to an inclined flat plate subject to sudden and ramp heating.International Journal of Thermal Sciences 49, 1600–1612.

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Gw = min(t/tw,1)×(q"/k)

X

Y

V

H g

U

Figure 1: Schematic of the computational domain and boundary conditions (See equation(42) for details)

Saha, S.C., Patterson, J.C., Lei, C., 2010c. Natural convection in attics sub-ject to instantaneous and ramp cooling boundary conditions. Energy Build-ings 42, 1192-1204.

Saha, S.C., Brown, R.J., Gu, Y.T. 2012. Scaling for the Prandtl numberof the natural convection boundary layer of an inclined flat plate underuniform surface heat flux. Internationa Journal of Heat and Fluid Flow,Under review.

Saha, S.C., Lei, C., Patterson, J.C. 2007. On the natural convection boundarylayer adjacent to an inclined flat plate subject to ramp heating, In Proceed-ing of the 16th Australasian Fluid Mechanics Conference, Crown Plaza,Gold Coast, Australia, 3-7 December, 121–124. (ISBN 978-1-864998-94-8).

Saha, S.C., Xu, F., Molla, M.M. 2011. Scaling analysis of the unsteady natu-ral convection boundary layer adjacent to an inclined plate for Pr > 1 fol-lowing instantaneous heating. ASME Journal of Heat Transfer 133, 112501.

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Figure 2: Snapshot of the isothermas and streamlines for Ra = 108 and Pr = 10 atdifferent times

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Figure 3: A schematic of the temperature and vertical velocity profiles on y = 0.5

Figure 4: (a) The plot of the computed data of velocity profiles calculated at y = 0.5 fortwo times for each of the 8 simulation cases for the case 0 < τ < τs, (b) scaled velocityprofiles plotted against the distance scaled by the distance from the plate to the velocitymaximum for each time.

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Figure 5: (a) The unscaled velocity profiles for two times for each of the simulation cases,(b) scaled velocity profiles plotted against the position scaled by the location of the velocitymaximum for the times in (a). The profiles are for the case τs < τ

Figure 6: (a) The unscaled velocity profiles at steady state for all simulation cases, (b)the velocity profiles at steady state scaled by the steady state maximum velocity plottedagainst the position scaled by the location of the velocity maximum.

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Figure 7: (a) The unscaled temperature profiles at steady state for all simulation cases.(b) The temperature profiles at steady state scaled by the final wall temperature plottedagainst position scaled by the steady state thermal boundary layer thickness

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Figure 8: Time series of the maximum vertical velocity in the boundary layer at y = 0.5 forall simulations; (a) computed velocities. (b) velocities scaled by vs plotted against τ/τs.

(c) velocities scaled by the steady state value Ra2/5Pr2/5

1+Pr1/2and plotted against (τ/τw)

2/5

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Figure 9: Time histories of the plate surface temperature for all simulations. (a) computedtemperature (b) scaled temperature plotted against scaled times

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