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A semi-implicit fractional step method immersed bound-

ary method for the numerical simulation of natural con-

vection non-Boussinesq ows

Dmitry Zviaga1, Ido Silverman2, Alexander Gelfgat3and Yuri Feldman1

1Department of Mechanical Engineering, Ben-Gurion University of the Negev,

P.O. Box 653, Beer-Sheva 84105, Israel.

2Soreq Nuclear Research Center, 81000 Yavne, Israel.

3School of Mechanical Engineering, Tel Aviv University, Tel Aviv 6997801,

Israel.

Abstract. The paper presents a novel pressure-corrected formulation of the immersed

boundary method (IBM) for the simulation of fully compressible non-Boussinesq natural

convection ows. The formulation incorporated into the pressure-based fractional step

approach facilitates simulation of the ows in the presence of an immersed body charac-

terized by a complex geometry. Here, we rst present extensive grid independence and

verication studies addressing incompressible pressure-driven ow in an extended chan-

nel and non-Boussinesq natural convection ow in a dierentially heated cavity. Next,

the steady-state non-Boussinesq natural convection ow developing in the presence of

hot cylinders of various diameters placed within a cold square cavity is thoroughly in-

vestigated. The obtained results are presented and analyzed in terms of the spatial

distribution of path lines and temperature elds and of heat ux values typical of the

hot cylinder and the cold cavity surfaces. Flow characteristics of multiple steady-state

solutions discovered for several congurations are presented and discussed in detail.

Key words: natural convection non-Boussinesq ows, pressure-corrected immersed boundary method,

multiple steady state solutions.

1 Introduction

The ability to accurately simulate natural convection ows is critical for a wide range of

engineering applications, including cooling electronic equipment, minimizing heat losses

in buildings, investigating atmospheric ows, modeling heat transfer, and preventing ac-

cidents in the nuclear industry, to name but a few. The methods typically utilized for

the simulation of natural convection ows can be classied into two major groups, one

relying on weakly compressible approximations and the other addressing the fully com-

pressible ow, as extensively reviewed in [1]. The rst group of methods treats the ow

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2

as incompressible. In these methods, the buoyancy is introduced by employing either

the Boussinesq approximation, which accounts for density variations in the gravity term

and may also account for variations in the thermophysical properties of the ow, or Gay-

Lussac-type approximations, which account for density variations in both the gravity and

advection terms. The second group uses fully compressible Navier Stokes (NS) and energy

equations.

The majority of weakly compressible approximations, which were developed for the

simulation of low-Mach-number compressible ows, are based on an asymptotic model.

The asymptotic model for the simulation of thermally driven natural convection ows

was formulated for the rst time in [2] and is known in the literature as the “classical

low-Mach-number model.” The key idea was to split the pressure eld into a large, time-

dependent thermodynamic part and a stationary part that includes extremely small spatial

deviations. Such a decomposition was found to be applicable for the simulation of low-

Mach-number thermally driven ows, as it provides the same order of magnitude for all

the terms in the momentum and energy equations and eliminates acoustic waves. Signi-

cant progress in this eld may be attributed to the works [3–5] and to the study [6], which

employed algorithms based on nite dierences and spectral methods, respectively, for the

simulation of non-Boussinesq natural convection conned ows. A weakly compressible

approximation was used in further simulations of numerous non-Boussinesq natural con-

vection ows [7–14] to address various problems in physics and computational science. The

common drawback of all weakly compressible approximations is that the results obtained

for ows with a dominant hydrodynamic part (e.g., high velocity ows) maybe inaccurate,

which means that a fully compressible approach must be used.

Fully compressible approximations for low-Mach-number ows typically employ either

density-based or pressure-based solvers. Although density-based formulations were tradi-

tionally utilized for simulating high-speed compressible ows (neglecting viscous eects),

eorts have been made to extend the applicability of these formulations to congurations

in which viscous eects play a signicant role. These congurations include laminar nat-

ural convection compressible ows [7, 15, 16], low-Mach-number injection ows [11], and

natural convection ows in a laminar-turbulent transition regime [17]. A key feature of

density-based solvers is that continuity, momentum, and all other transport equations are

rst solved in a fully coupled manner, and the pressure eld is then derived from an equa-

tion of state. As a result, the coupled operator is typically ill conditioned when applied

to low-Mach-number ows, whose treatment requires sophisticated numerical techniques,

such as preconditioning and dual-time stepping. When the simulations are performed on

high-resolution grids or applied to 3D ows, density-based solvers typically suer from

a slow convergence rate and produce results with low accuracy. Pressure-based solvers

oer a better alternative for the simulation of fully compressible natural convection ows.

In pressure-based solvers, the calculation of the pressure is separated from that of the

velocity eld. In the rst step, the pressure eld is taken from the previous time step

and the momentum equations are solved, thereby yielding the velocity eld, which does

not satisfy the continuity equation. In the second stage, a pressure-correction equation

3

(which is derived from the continuity equation) is solved. Thereafter, the pressure and the

velocity components are corrected to meet the continuity constraint. Finally, the solution

of the pressure-correction equation is followed by the solution of the energy equation and

updating of the density and the viscosity elds by relying on the equations of state and

Sutherland equations. Typically, outer iterations are needed to achieve convergence of the

velocity, the temperature, and the density elds and to satisfy the continuity constraint.

Despite the fact that the pressure-based approach (employing either fractional-step or

SIMPLE methods) is well documented in the literature [18, 19], numerical simulations

of fully compressible natural convection in enclosures performed by utilizing the above

algorithm are quite scarce, with the prominent exceptions of the studies of Sewall and

Tafti [20] and Barrios-Pina et al. [21]. In [20], a variable property algorithm based on the

fractional-step method was developed to simulate transient natural convection ows in the

presence of large temperature dierences without using the low-Mach-number assumption.

The algorithm was then adopted in [21] to conduct a thermodynamic analysis aimed at

determining the contribution of each term in the total energy equation.

In practice, most realistic engineering applications involving conned natural convec-

tion ow are characterized by complex geometries, which can signicantly challenge the

accuracy of the calculated ow properties. For these applications, the immersed boundary

method (IBM), initially developed by Peskin [22], may be used as a convenient tool to

simulate the ow in the presence of complex boundaries while maintaining an acceptable

level of accuracy. The IBM can simulate ow around complex, movable, and deformable

boundaries. The simulations take advantage of solvers utilizing compact and simple sten-

cils of discretized dierential operators that can be eciently employed on structured grids

or solvers based on the Lattice Boltzmann method (see e.g [23]). No-slip boundary condi-

tions and prescribed values of the temperature (or heat ux) on each immersed boundary

are enforced by introducing forces and heat uxes as additional unknowns in the problem.

Closure of the overall system is achieved by including additional equations in the form of

kinematic constraints for all the unknowns.

In the past decade, the IBM has been widely utilized for investigating natural con-

vection within enclosures with embedded discrete thermally active sources (or sinks) of

various geometries. Interest in this eld was generated by its relevance to a broad spectrum

of engineering applications based on gas-solid heat exchangers and a need to investigate

the instability characteristics of highly separated conned ows. Worth mentioning in this

context are the works [24–27] and the studies [28–32] that addressed natural convection

conned ows in the presence of bodies of complex two- and three-dimensional geometries,

respectively.

Studies utilizing the IBM for the analysis of thermal compressible ows are relatively

scarce. Most of the works in this eld either address high-Mach-number compressible ows,

focusing on transonic/supersonic transitions, or compare the characteristics of subsonic

ows with those of supersonic ows. For high-Mach-number ows, the impacts of viscosity

and the thermal behavior of the ow are negligible compared to the compressibility eects,

and thus both phenomena have typically been neglected when simulating high-Mach-

4

number ows. In contrast, in the simulation of low-Mach-number thermally driven ows –

the focus of the current study – both eects play a signicant role and should be carefully

addressed.

It should be noted that an accurate implementation of IBM forcing in low-Mach-

number compressible ows is still the subject of active research. In a recently developed

IBM scheme [33], Riahi et al. introduced a novel pressure-based correction of IBM forcing

(in addition to the classical one based on the time derivative of velocity)∗and applied

it to the analysis of three-dimensional low- and high-Mach-number pressure driven ows.

Comparison of the obtained results with the corresponding data obtained by body-tted

numerical simulations revealed that pressure correction of IBM forcing signicantly im-

proved the accuracy of the IBM procedure for low-Mach-number ows. An additional con-

tribution to the IBM for the simulation of compressible thermally driven conned ows

was made by Kumar and Natarajan [33], who developed a diuse immersed boundary

method for thermally driven non-Boussinesq ows. However, the method relies on a low-

Mach-number approximation that treats the governing equations as quasi-incompressible

and therefore cannot be considered as a fully compressible approach.

The present study thus aims to develop and extensively verify a general transient

pressure-based formulation for the simulation of thermally driven non-Boussinesq ows

within complex geometries. The simulations are performed by utilizing second-order back-

ward nite dierence and standard second-order nite volume methods [35] for the tem-

poral and spatial discretizations, respectively. To the best of our knowledge, the current

work is a rst of its kind in which the pressure-based correction IBM, originally developed

for compressible pressure driven ows [33], has been successfully incorporated into a fully

compressible pressure-based natural convection solver utilizing a semi-implicit fractional-

step algorithm.

2 Theoretical background

2.1 Physical model

Natural non-Boussinesq convection within a rectangular cavity of dimensions L×Hlled

with an ideal gas is considered. In the presence of a gravitational eld −gˆ

j, the ow is

driven by the temperature dierence between the cold boundaries of the cavity and the

hot surface of a cylinder located at the geometrical center of the cavity, as shown in Fig.1.

The left (LW), right (RW), top(TW) and bottom (BW) walls of the cavity are maintained

at a constant cold temperature Tc, while the surface of the cylinder (CL) is maintained

at a constant hot temperature Th. For the most general case in which the temperature

dierence Th−Tcis not restricted to small values, the natural convection ow generated

within the cavity is fully compressible.

The ow is governed by a set of non-dimensional continuity, momentum, and energy

∗We note in passing that pressure correction of the direct forcing IBM must not be confused with a

pressure-correction equation of fractional-step- and the SIMPLE-related methods.

5

Figure 1: Geometry and boundary conditions for a cold cavity with a hot cylinder at the center.

LW

TW

RW

BW

CL

equations:

∂ρ

∂t+∇·(ρ

u) = 0, (2.1)

∂(ρ

u)

∂t+∇·(ρ

u⊗

u)=−RaPr

2ερˆ

j−1

κM2

0

∇p+Pr∇·µ(∇

u+∇

uT)−2

3(∇·

u)I+

fΓ,(2.2)

∂(ρCpT)

∂t+∇·(ρ

uCpT) = ∇ ·(k∇T) + κ−1

κ∂p

∂t+

u·∇p+qΓ,(2.3)

ρ=p

T=Cρp.(2.4)

where

u(u,v),p,T,ρ,µ,k,Cp,κ,Cρare the non-dimensional velocity, pressure, temperature,

density, dynamic viscosity, thermal conductivity, specic heat capacity at constant pres-

sure, ratio of specic heat capacities and density coecient, respectively, and Iis the

unity matrix. The impact of the immersed cylinder on the surrounding ow is addressed

by introducing the source terms

fΓ,qΓcorresponding to the volumetric force and heat

source, respectively. Note that in accordance with the IBM formalism as detailed in

section 2.2, Eqs. (2.1-2.4) are solved on the whole computational domain including the

cylinder interior.

Eqs. (2.1-2.4) are rendered dimensionless using the characteristic scales L0for length,

L2

0/α0for time, α0/L0for velocity, p0for pressure, ρ0for density, T0for temperature, Cp0

for specic heat capacity at a constant pressure, µ0for dynamic viscosity, k0for thermal

conductivity and α0is thermal diusivity. The dimensionless groups governing the ow

under consideration are the Rayleigh number (Ra), the Prandtl number (Pr), the Mach

number (M0)and the normalized temperature dierence parameter (ε)dened as:

Ra =ρ0g(Th−Tc)L3

0

µ0α0T0

,Pr =µ0

ρ0α0

,M2

0=α2

0/L2

0

κRT0

,ε=Th−Tc

Th+Tc

=Th−Tc

2T0

(2.5)

6

Note that despite the fact that the value Th−Tcis included in the denition of Ra

and ε, the two parameters are independent of each other and can be changed separately.

The Sutherland law determining the dependence of both the viscosity and the thermal

conductivity values on temperature is applied, giving:

µ=1+Cµ

T+Cµ

T3/2,(2.6)

k=1+Ck

T+Ck

T3/2,(2.7)

where Cµ,Ckare the Sutherland non-dimensional temperatures for viscosity and thermal

conductivity included in the governing Eqs. (2.2-2.3).

2.2 IBM for thermal compressible ow

To enforce the kinematic constraints of no-slip and of the prescribed temperature value

on the surface of the embedded cylinder, the IBM is employed when solving momentum

and energy equations. To impose the kinematic constraints, the IBM utilizes regular

(typically Cartesian) Eulerian grids by introducing a set of additional volumetric forces

and heat sources on the surface of the immersed body, which is described by a discrete

set of Lagrangian points. The current study extends the pressure-corrected direct forcing

IBM presented in [33] – originally developed for compressible isothermal ows – to the

simulation of non-Boussinesq natural convection ows. Similarly to the conventional direct

forcing IBM, the formulation developed here incorporates interpolation and regularization

operators facilitating an exchange of data between Eulerian and Lagrangian grids and

a procedure enabling the direct calculation of the Lagrangian volumetric heat and force

sources.

2.2.1 Interpolation

The interpolation step transfers quantities (e.g., (ρ

u),(ρCpT),(∇p)) from the Eulerian

mesh to the points determining the Lagrangian surface ∂B. The procedure employs an

interpolation operator Idened as:

I[uE+vE]

XL=

UL+

VL=∑

i∈Nxj∈Ny

(

UL+

VL)Nx,Ny

i,jδ(xi−XL)δ(yj−YL)∆x∆y(2.8)

where uE(ρ

u,ρCpT),vE(∇p,0)are physical properties calculated on a Eulerian Nx×Ny

grid, while

UL(ρL

UL,ρLCpLTL),

VL(∇PL,0)are the corresponding counterparts calculated

on a Lagrangian grid, and δis a discrete Dirac delta function dened below. Note that we

used a bold vector notation to distinguish between the interpolated terms yielded by the

solution of momentum or energy equation. From the dierent smeared approximations

of the delta functions, we chose the function described by Roma et al. [36], which was

7

specically designed for use on staggered grids, where even/odd de-coupling does not

occur. This approximation is expressed as:

δ(r) =

1

3∆r1+−3|r|

∆r2

+1for |r| ≤ 0.5∆r,

1

6∆r5−3|r|

∆r−−31−|r|

∆r2

+1for 0.5∆r≤ |r| ≤ 1.5∆r,

0otherwise,

(2.9)

where ∆ris the cell width in the rdirection. It is noteworthy that the chosen discrete

delta function is supported over three cells. No signicant dierences in the results are to

be expected if other discrete delta functions are used [37].

2.2.2 Direct forcing

After completing the interpolation step, the Lagrangian volumetric forces and heat sources

are calculated as suggested in [33]:

FL=1

∆t(

Ud

L−

UL)−(

Vd

L−

VL)·ˆnns (2.10)

where

FL(

FL,QL)is a direct forcing term consisting of the volumetric force and heat sources,

respectively, the superscript ddenotes the kinematic constraints imposed on the surface of

the immersed body and ˆnns is a unit vector in the direction perpendicular to the surface

of the immersed body.

2.2.3 Regularization

The regularization step smears the values of the volumetric sources calculated at the

Lagrangian points back to the Eulerian grid. The procedure is implemented by utilizing

the same delta functions as in the interpolation step. The values of the volumetric force

and heat source terms evaluated on the Eulerian mesh by utilizing the regularization

operator Rare given by:

fΓ(x,y)=R[

FL]E=∑

L∈NL

FLδ(xi−XL)δ(yj−YL)∆x∆y(2.11)

where

fΓ(x,y)(

fΓ(x,y),qΓ(x,y)) is the Eulerian direct forcing term consisting of the volu-

metric force and heat sources.

3 Numerical methodology

The ow is governed by a system of continuity, momentum, and energy equations, which

are solved numerically. This study utilizes the fractional-step method, which separates

8

the calculation of the pressure eld from the calculation of the velocity eld at each time

instance. When utilized in the context of the Boussinesq approximation (i.e., the ow is

assumed to be incompressible), the method consists of a number of basic steps employed

at each time instance: (i) the predictor step aimed at the estimation of the non-solenoidal

velocity eld by utilizing the pressure eld from the previous time step; (ii) the corrector

step aimed to obtain the pressure correction for the current time step; (iii) the projection

step using the pressure-correction values to update the pressure eld and to project the

velocity eld on the divergence free subspace; and (iv) solution of the energy equation.

For non-Boussinesq (i.e., fully compressible) natural convection ows, the procedure is

more complicated [18], since in that case the pressure constitutes a thermodynamic prop-

erty rather than a simplied hydrodynamic property. An additional diculty is that for

compressible ows under realistic conditions, density, viscosity, and thermal conductivity

are not constant. The ow around a body of complex geometry is resolved by employing

the direct forcing IBM applied to compressible ow, and the velocity, pressure, and tem-

perature kinematic constraints to be enforced on the surface of the immersed body. The

methodology and the numerical formulation incorporating a pressure-corrected IBM into

a semi-implicit fractional-step method developed for the simulation of natural convection

ow around complex geometries is presented below.

3.1 Computational procedure

In this section the details on implementation of fractional step method to satisfy the

continuity equation and the IBM formalism to satisfy the kinetic constraints on the syrface

of the immersed body are given. Note that the second order backward nite dierence

and standard second order nite volume method [35] was used for the temporal and the

spatial discretizations, respectively.

3.1.1 Predictor step

3ρm−1

u∗

2∆t−Pr∇·µm−1(∇

u+∇

uT)−2

3(∇·

u)I∗/n

=

=−RaPr

2ερm−1ˆ

j−N(ρm−1,

un)−1

κM2

0

∇pm−1+4(ρ

u)n−(ρ

u)n−1

2∆t,

(3.1)

where

u∗is the predicted velocity that has to be calculated on the basis of the velocity eld

un, known from the previous time step, and on the basis of the density, dynamic viscosity

and pressure elds known from the previous outer iteration m−1. The purpose of the

outer iteration is to impose the continuity constraint on the predicted velocity eld

u∗at

the end of the iteration process. The term N(ρm−1,

un)appearing in the right hand side of

Eq. (3.1) denotes the non-linear convective terms, so that Eq. (3.1) is solved sequentially

for each velocity component. For this reason the second term of the left hand side of

Eq.(3.1) contains both terms that are treated implicitly (denoted by * superscript) and

explicitly (i.e. taken from the nth time step). This notation was introduced to distinguish

9

between velocity components corresponding to dierent directions. That is, the velocity

components coinciding with the direction of the corresponding momentum equation are

treated implicitly while the velocity components perpendicular to this direction are taken

from the previous time step. Note also that for incompressible and Boussinesq ows

with constant viscosity, the second term may be simplied to Pr∇2(

u∗)and treated fully

implicitly.

3.1.2 First momentum Corrector

After obtaining the predicted velocity, the pressure-correction equation, which has been

derived from the continuity equation, is solved as follows:

3

2∆tCρp′−∇· 2∆t

3κM2

0

∇p′−∇· (Cρp′

u∗) = −∆˙

m∗,(3.2)

where ∆˙

m∗is the mass ow imbalance generated because the predicted velocity does not

necessarily satisfy the continuity equation:

∆˙

m∗=3ρm−1−4ρn+ρn−1

2∆t+∇·(ρm−1

u∗).(3.3)

Solution of the pressure-correction equation yields the pressure correction eld p′used for

calculation of the intermediate pressure:

p∗=pn+p′,(3.4)

which is subsequently used as a predictor after the presence of the immersed body has

been taken into account.

3.1.3 Application of the IBM for velocity to enforce the non-slip kinematic constraint on

the surface of the immersed body

After acquiring the intermediate pressure, the IBM for velocity is implemented via Eqs.

(2.8-2.11). Note that the term

fΓis not recalculated during the outer iteration and is

determined only once at the beginning of the time step:

fΓ=Rρm−1

L

∆t(

ud

L−

u∗

L)−(∇pd

L−∇p∗

L)·ˆnns E

.(3.5)

As a result, the calculated volumetric Eulerian force exerted by the surface of the immersed

body is added to the right-hand side of the momentum equation.

3.1.4 Solution of the momentum equation with the impact of the immersed body

After the Eulerian force has been calculated, the momentum equation with an updated

right-hand side is solved again to determine the new velocity eld that takes into account

10

the impact of the immersed body:

3ρm−1

um

2∆t−Pr∇·µm−1(∇

u+∇

uT)−2

3(∇·

u)Im/n

=

=−RaPr

2ερm−1ˆ

j−N(ρm−1,

un)−1

κM2

0

∇p∗+4(ρ

u)n−(ρ

u)n−1

2∆t+

fΓ.

(3.6)

Similarly to Eq. (3.1), the above equation is solved sequentially for each velocity compo-

nent. For this reason the second term of the left hand side of Eq.(3.6) contains both terms

that are treated implicitly (i.e. the current iteration terms denoted by msuperscript) and

explicitly (i.e. taken from nth time step). This notation was introduced to distinguish

between velocity components corresponding to dierent directions. That is, the veloc-

ity components coinciding with the direction of the corresponding momentum equation

are taken from the current iteration and treated implicitly while the velocity components

perpendicular to this direction are taken from the previous time step.

3.1.5 Second momentum corrector

At this stage, the pressure-correction equation with the updated right-hand side is solved:

3

2∆tCρp′′ −∇·2∆t

3κM2

0

∇p′′−∇· (Cρp′′

um) = −∆˙

mm,(3.7)

where ∆˙

mmis a mass ow imbalance that arises because the predicted velocity still does

not necessarily satisfy the continuity equation, where ∆˙

mmis given as:

∆˙

mm=3ρm−1−4ρn+ρn−1

2∆t+∇·(ρm−1

um).(3.8)

Next, the new pressure and the intermediate density elds are calculated by:

pm=p∗+p′′,ρ∗=ρn+Cρ(p′+p′′).(3.9)

3.1.6 Solution of the energy equation without the impact of the immersed body

We next solve the energy equation:

3ρ∗CpT∗

2∆t+∇·(ρ∗

umCpT∗)−∇· (km−1∇T∗) =

=κ−1

κ3pm−4pn+pn−1

2∆t+

um·∇pm+4ρnCpTn−ρn−1CpTn−1

2∆t.

(3.10)

Note, the T∗eld constitutes the intermediate temperature, which was obtained without

considering the presence of the immersed body.

11

3.1.7 Application of the IBM to enforce a given temperature on the surface of the im-

mersed body

At each time step in the rst correction iteration, the IBM for temperature is implemented

via Eqs. (2.8-2.11) after acquiring the intermediate temperature. Note that the term qΓis

not recalculated during the outer iteration and is determined only once at the beginning

of the time step:

qΓ=Rρ∗

LCp

∆t(Td

L−T∗

L)E

.(3.11)

As a result, the calculated volumetric Eulerian heat source exerted by the surface of the

immersed body is added to the right-hand side of the energy equation.

3.1.8 Solution of the energy equation with the impact of the immersed body

We next solve the energy equation:

3ρ∗CpTm

2∆t+∇·(ρ∗

umCpTm)−∇· (km−1∇Tm) =

=κ−1

κ3pm−4pn+pn−1

2∆t+

um·∇pm+4ρnCpTn+ρn−1CpTn−1

2∆t+qΓ.

(3.12)

3.1.9 Updating the thermophysical properties

After the energy equation has been solved, the viscosity µm

ij , the thermal conductivity

km

ij , the coecient Cρ, and the density ρm

ij are updated by using the Sutherland equations

(2.6),(2.7) and the equation of state (2.4).

3.1.10 The outer iteration loop

The general formulation of the fully compressible semi-implicit fractional-step method

with the embedded IBM governed by Eqs. (3.2),(3.4-3.12) constitutes the outer iteration

loop that is employed at each time step. At the end of the iteration, after updating the

thermophysical properties, the solver performs a mass conservation check based on the

value of the L-innity norm calculated for the dierence between the values of p,ρand

u

elds in the current and previous iterations. The outer iteration terminates after the value

of 10−6of the L-innity norm has been reached for each eld and the simulation proceeds to

the next time step by assigning

un+1=

um,pn+1=pm,Tn+1=Tmand ρn+1=ρm. To sum up,

we present a owchart (see Fig. 2 summarizing the sequence of stepst hat must be taken

to calculate all the ow elds at a given time step with and without an immersed body.

It can be seen that the ow simulation involving an immersed body requires additional

steps to explicitly calculate the volumetric forces and heat uxes necessary to satisfy the

kinematic constraints on the surface of the body.

12

Figure 2: Block diagram of the algorithmic sequence required to calculate all the ow elds at a given time

step: to the left – without immersed body; to the right – with immersed body.

Predictor step - ݑ

כ

Momentum corrector - ǡݑǡߩProjection step -

ǡݑ

ǡߩ

כ

Solution of the energy equation - ܶ

Updating thermophysical properties -

ߤ

ǡ݇

ǡߩ

Checking continuity constraint

Proceed to the next time step

Continuity conserved

Continuity

does not

conserved

Predictor step - ݑכ

First momentum corrector -ᇱǡכIBM for velocity - ݂ԦSolution of momentum equation with the impact of the immersed body -ݑUpdating thermophysical properties - ߤǡ݇ǡߩChecking continuity constraint

Proceed to the next time step

Continuity conserved

Continuity

does not

conserved

Second momentum corrector -ᇳǡǡߩכSolution of energy equation without the impact of the immersed body -ܶכIBM for temperature -ݍSolution of energy equation with the impact of the immersed body - ܶ

3.2 Solution of the discretized momentum, pressure-correction and energy equa-

tions

As a result of the non-constant density ρand the dependence of the dynamic viscosity µ

and the conduction coecient kon the temperature typical of the fully compressible ow,

the discretized momentum, pressure correction, and energy equations contain time varying

coecients. Therefore, the strategies for reaching an ecient solution of the discretized

equations should be chosen with care. In the current study, two dierent strategies, one

based on an iterative solution and the other based on a direct solution of the discretized

governing equations, were investigated.

The iterative solution utilized the bi-conjugate gradient stabilized (BiCGstab) method

[38]. The key idea was to treat only the linear terms of the Helmholtz operator implicitly,

while all the non-linear terms were taken either from the previous time step or from the

previous iteration. Exploiting the fact that for a 2D problem the Helmholtz operator is

built up of 5 non-zero diagonals, we eciently implemented its product by an arbitrary

vector, constituting a major part of the BiCGstab algorithm. However, as is the case for

many other iterative methods, the BiCGstab converges up to a certain accuracy, after

which it saturates, so that no further decrease of residuals is possible. This limitation can

slow down the convergence of the outer iterations, and we therefore sought to eliminate

this possible problem by replacing the BiCGstab by the direct method proposed by Lynch

et al. [39]. This approach, designated tensor product factorization (TPF) in [39], is based

on eigenvalue decompositions (EVDs) of one-dimensional derivative operators and can be

applied for the direct inverse of the Helmholtz operators in rectangular domains. Vitoshkin

13

& Gelfgat [40] demonstrated the computational eciency of the TPF method for 2D and

3D natural convection benchmark problems by applying the Boussinesq approximation

and showed that for ne grids and large Grashof numbers it yields computationally faster

time steps than BiCGstab or multigrid iterations. Additionally, it can easily be observed

that after application of the TPF solver the relative residual remains below 10−12.

Unfortunately, the EVD method cannot be implemented directly into the correction

equations (3.2) and (3.7), because the coecients of the Helmholtz-like dierential operator

are not constants. Thus, to apply the EVD solver, all the governing equations should be

reformulated. For example, Eq. (3.7) is modied into:

3

2∆tCρp′′,m−∇·2∆t

3κM2

0

∇p′′,m=−∆˙

mm+∇·(Cρp′′,m−1

u∗)−3

2∆t(Cρ−Cρ)p′′,m−1,(3.13)

where Cρ=(Cρ)min +(Cρ)ma x/2 and mis the number of the outer iteration. In this

formulation, the left-hand side of Eq. (3.13) is a Helmholtz-like elliptic operator with

constant coecients, whose discretized inverse can be expressed via a one-dimensional

EVD of the second-derivative operators [41].

4 Verication study

The methodology described in Section 2 was veried by applying it to the solution of two

benchmark problems—simulating incompressible and thermally driven compressible ows,

i.e., ows driven by two dierent mechanisms. For the verication, we chose to simulate

the isothermal ow in a long narrow rectangular channel driven by a pressure gradient

and the natural convection ow within a dierentially heated square cavity driven by a

temperature gradient. The results obtained for the two congurations were compared with

the data available in the literature.

4.1 Test case I - Isothermal compressible ow in a narrow channel

4.1.1 Test case overview

This test case examines the capability to carefully address incompressible isothermal ow

by applying the currently developed methodology for compressible ow. The ow within

an extended channel was selected as a computational testbed to minimize the eect of the

outow boundary on the upstream recirculation zones. The schematics of the geometric

properties and the boundary conditions of the conguration under consideration are shown

in Fig. 3.

The uid enters the domain at the upper half of the left side, proceeds through the

channel, and exits at the right side. No-slip velocity and zero-gradient pressure boundary

conditions are applied on all the rigid walls. At the inlet, the vertical component of the

velocity is equal to zero; the zero-gradient boundary condition is applied to the pressure

14

Figure 3: Schematics of the ow within an extended channel.

eld; and a parabolic distribution with maximal and average values equal to umax =1.5

and uavg =1, respectively, is assigned to the horizontal velocity component. At the outlet,

zero values are set for the normal stress, τx x =−p+µ∂u/∂x, and for the vertical velocity

component. Considering that the pressure eld at the outlet is known and is set to zero,

the gradient of the horizontal velocity component is also equal to zero.

The non-dimensional continuity and momentum equations and the equation of state

governing the ow under consideration are:

∂ρ

∂t+∇·(ρ

u) = 0, (4.1)

∂(ρ

u)

∂t+∇·(ρ

u⊗

u) = −∇p+1

Re ∇·µ(∇

u+∇

uT)−2

3(∇

u)I,(4.2)

ρ=κM2

0

T0

p,(4.3)

where the Reynolds number, Re is based on the average velocity.

4.1.2 Test results and comparison with a benchmark in the literature

The results obtained by the developed methodology were compared to the corresponding

data provided in [42] for the value of Re =800. Figs. 4-7 present a comparison between the

contours of the currently obtained ow elds and the corresponding results reported in [42],

which serves as the benchmark for this comparison. As can be seen from Figs. 4-7, the

values of all the currently obtained quantities lie in the same range as the corresponding

data reported in [42]. The distribution of the streamlines, shown in Fig. 4, along with

the distributions of the vorticity and velocity magnitude elds, shown in Figs. 6 and 7,

respectively, conrm the presence of staggered low-speed vortices adjacent to the upper

and lower walls. The pressure eld, shown in Fig. 5, conrms the presence of a “pressure

pocket” adjacent to the bottom wall between x=6and x=7. As can be seen clearly from

Fig. 8, the density variations over the entire domain are insignicant, and the ow can

safely be considered as incompressible.

15

Figure 4: Comparison between the distributions of the streamlines in (a) the current study and (b) the benchmark

study [42]. Grid 3000×100.

Figure 5: Comparison between the pressure distribution in (a) the current study and (b) the benchmark study

[42]. Grid 3000×100.

16

Figure 6: Comparison between the distribution of vorticity in (a) the current study and (b) the benchmark

study [42]. Grid 3000×100.

Figure 7: Comparison between the distribution of the velocity magnitude in (a) the current study and (b) the

benchmark study [42]. Grid 3000×100.

Figure 8: Distribution of the density eld. Grid 3000×100.

17

Figs. 9-10 present a comparison between the currently obtained ow characteristics

and the corresponding data reported in [42]. A comparison of the pressure and the shear

stress distributions along the upper and lower walls of the channel is shown in Fig. 9a

and 9b, respectively. A comparison of the currently obtained and benchmark distributions

of the horizontal and vertical velocity components, pressure, vorticity, horizontal velocity

gradient and normal stress along two vertical lines passing through x=7and x=5is

presented in Figure 10. All the gures demonstrate the same trends for all the ow

characteristics obtained in the current and benchmark studies.

Figs. 9a, 10c and 10f compare the corresponding pressure and the normal stress elds

and show excellent agreement between the current and the benchmark values for the two

quantities. Acceptable agreement was also obtained for all the other ow characteristics,

including the values of both velocity components, the vorticity, and the gradient of the

horizontal component of the velocity. An absolute deviation between the values of the

above characteristics is limited to 1percent for the entire range of y coordinates with the

exception of the regions in which extremums of the vertical velocity and the gradient of

the horizontal velocity are observed. In these regions, the absolute deviation between the

current and the corresponding benchmark results is limited to 8percent.

In summary, the acceptable agreement between the currently obtained and benchmark

results for the entire range of ow characteristics veries the suitability of our numerical

methodology for the simulation of almost incompressible ows.

4.2 Test case II - Natural convection ow in a dierentially heated cavity

4.2.1 Test case overview

The results presented in this section were obtained by applying the developed method-

ology to the simulation of compressible natural convection ow in a dierentially heated

square cavity. The ow is driven by the temperature dierence between two vertical walls

under the inuence of gravity. The obtained ow characteristics were compared with

the corresponding independently obtained data [4] that served as the benchmark for this

part of our study. The governing equations, characteristic values, and non-dimensional

parameters of this ow conguration are presented in Section 2.

The hot and cold walls of the cavity are maintained at constant temperature values

of Th=1+εand Th=1−ε, respectively, and are thermally insulated. No-slip and zero

gradient boundary conditions are applied for all the velocity components and the pres-

sure, respectively, on all the cavity walls. The schematics summarizing the geometry and

boundary conditions of this ow conguration are shown in Fig. 11.

4.2.2 Test results and comparison with a benchmark in the literature

The results obtained in the current study were compared with the corresponding bench-

mark data reported in [4]. The comparison focused on the distributions of the velocity

and the temperature. The data was compared in terms of the values of the vertical tem-

perature stratication parameter θAand the Nusselt number determined by Eqs. (4.4)

18

Figure 9: Distribution of (a) pressure and (b) shear stress elds along upper and lower channel walls. Grid

3000×100.

19

Figure 10: Distribution of: (a) horizontal velocity component; (b) vertical velocity component; (c) pressure;

(d) vorticity; (e) horizontal velocity gradient; and (f) normal stress along vertical lines passing through the x=7

and x=15 coordinates. Grid 3000×100.

20

Figure 11: Geometry and boundary conditions of the dierentially heated cavity.

RW

LW

TW

BW

and (4.5), respectively. In accordance with [4], θAwas calculated by:

θA=AR

2εdT

dy x=1

2,y=1

2

,(4.4)

where AR =H/Lis the aspect ratio.

The average Nusselt number was determined as:

Nu =−1

SS

k∇T

2εdS,(4.5)

where ∇Tis the temperature gradient at the wall, and Sis the non-dimensional surface

area.

In the current study, the Nusselt number was calculated for the hot wall of the cavity.

The calculations were performed for the range of Ra ∈[103,107]and ε=0.005,0.2,0.4 and

0.6. In all the simulations, the value of the aspect ratio AR=H/Lwas equal to unity. Figs.

12-13 summarize the comparison between our calculations and the benchmark values [4]

for the velocity and the temperature elds obtained for the value of Ra=105and the entire

range of εvalues. Good agreement between the two sets of values can clearly be seen for

all the simulations. Note that the temperature and velocity distributions vary with ε. For

the lowest value of ε, i.e., ε=0.005, both the velocity and the temperature distributions

are almost skew-symmetric relative to the cavity center and resemble the distributions

21

Figure 12: Comparison of the contours for ε=0.005,Ra=105: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

Figure 13: Comparison of the contours for ε=0.2,Ra =105: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

typical of incompressible ows obtained by applying the Boussinesq approximation. With

increasing εvalues, the ow loses its skew-symmetry as a result of the dependence of its

conductivity kand viscosity µon temperature.

A similar comparison was made for εconstant at its highest value, ε=0.6, and varying

Ra over the entire range of Ra values, as shown in Figures 14-19. Good agreement was

obtained between our values and the benchmark results [4]. It can clearly be seen that

under these conditions the ow is characterized by a breaking of the skew-symmetry

relative to the cavity center, even at the lowest value of R a =103. For the higher Ra ow

regimes dominated by convective heat transfer, the breaking of skew-symmetry becomes

Figure 14: Comparison of the contours for ε=0.4,Ra =105: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

22

Figure 15: Comparison of the contours for ε=0.6,Ra =105: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

Figure 16: Comparison of the contours for ε=0.6,Ra =103: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

much more pronounced. The breaking of skew-symmetry may be conrmed by examining

the dierences in the thickness of the boundary layers developing near the hot and cold

walls. In fact, the dynamic viscosity of the ideal gas increases with temperature, leading

to a thickening of the boundary layer in the vicinity of the hot wall of the cylinder. At the

same time, the thickness of the boundary layer decreases close to the cold vertical wall,

which, again, is a consequence of a local decrease in viscosity values as a result of the lower

temperatures prevailing in this region.

Figs. 20 and 21 present a comparison of our results with the benchmark values [4]

for the vertical temperature stratication parameter and the averaged Nusselt number

Figure 17: Comparison of the contours for ε=0.6,Ra =104: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

23

Figure 18: Comparison of the contours for ε=0.6,Ra =105: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

Figure 19: Comparison of the contours for ε=0.6,Ra =106: (a) velocity, current study, (b) velocity, benchmark

study [4], (c) temperature current study, (d) temperature, benchmark study [4].

versus Ra values. The simulations were performed on four dierent grids, namely, two

uniform grids of 100×100 and of 200×200 cells, and two non-uniform grids of 100×100

and of 200×200 cells stretched toward the cavity walls to accurately resolve the thinnest

boundary layers.

As can be seen from Fig. 20, there was good agreement between our results and the

benchmark results [4] for the vertical temperature stratication parameter. Of partic-

ular interest was the nding that stretching the grid toward the cavity boundaries was

not always the optimal way to increase the accuracy of the results when the values for

comparison were acquired close to the cavity center, as can be seen from the θAvalues

obtained on a stretched grid with ε=0.005 and Ra=106or a stretched grid with ε=0.2 and

Ra =103. At the same time, asymptotic convergence of the θAvalues with increasing the

grid resolution was clearly observed for both stretched and uniform grids. Fig. 21 demon-

strates that the Nu values obtained on the uniform and stretched grids built of 200×200

and 100×100 cells, respectively, showed better agreement with the benchmark results

than the corresponding Nu values obtained on the uniform grid built of 100×100 cells,

especially for high Rayleigh numbers. Thus, stretching the grid toward the cavity bound-

aries is the condition of choice when analyzing characteristics based on the temperature

gradients at the cavity boundaries.

In summary, the acceptable agreement between our results and those of the benchmark

study [4] for the entire range of operating conditions and ow characteristics successfully

veries the suitability of our numerical methodology for the simulation of compressible

24

Figure 20: Vertical temperature stratication parameter vs. Rayleigh number: comparison of our results with

benchmark values [4].

natural convection conned ows.

5 Results and discussion

This section presents the results of a parametric study performed to simulate the natural

convection ow developing from the hot surface of a cylinder placed within a square cold

cavity (see Fig. 1). The results were obtained for a wide range of governing parameters:

Ra∈{103,104, 105,106},ε∈{0.005,0.2,0.4,0.6}and R/L∈{0.1,0.2,0.3,0.4}. The simulations

were performed on two uniform grids having 100 or 200 nodes in each direction, with the

time steps of ∆t=10−7and ∆t=10−8, respectively.

The obtained results illustrate changes of the velocity and temperature elds when

the temperature-dierence parameter εwas varied between 0.005 and 0.6 for the same Ra

and R/Lvalues. Additionally, the results were approximated by Nu−Ra power law ts

calculated by employing the least-squares technique. The Nusselt number Nu was calcu-

lated as is detailed below in subsection 5.2. The Nusselt numbers obtained for the whole

range of Ra and for the lowest value of the temperature-dierence parameter (ε=0.005)

were compared with the corresponding results available in literature [28], [43] calculated

by employing the Boussinesq approximation.

25

Figure 21: Nusselt number vs. Rayleigh number: comparison of our results with benchmark values [4].

5.1 Qualitative observations

Figs. 22-37 summarize the results in terms of the spatial distribution of the ow path lines

and the temperature elds obtained for the values of ε=0.005 and ε=0.6 for the entire range

of Ra and R/Lvalues on a uniform grid of 200 cells in each direction. The purpose of this

part of the study was to investigate dierences between the ow characteristics typical

of the lowest (ε=0.005) and the highest (ε=0.6) values of the temperature-dierence

parameter.

Figs. 22-23 and Figs. 30-31 show that for the lowest Rayleigh number, Ra=103, there

were no signicant dierences between the spatial distributions of the path lines and the

temperature elds, respectively, obtained for the lowest (ε=0.005) and the highest (ε=0.6)

values of the temperature-dierence parameter, regardless of the cylinder’s diameter. The

congurations for this Ra number did not contain secondary convective cells, and the

temperature distribution was close to linear along the radial direction from the hot cylinder

to the cold cavity walls, as may be expected for systems in which conduction constitutes the

major heat transfer mechanism. As the Ra value increased, the dierences became more

visible. Figs. 24-25 and Figs. 32-33, in which the path line and temperature distributions,

respectively, are shown for Ra =104, showed that the dierences in path lines remained

insignicant (with no secondary convective cells), but the temperature distribution for the

highest value of εwas slightly shifted upwards compared to that observed for the lowest ε

value. As for the two highest values of Ra =105and Ra =106, the dierences between the

path line and temperature distributions corresponding to the congurations characterized

by the two limit values of εcould be clearly recognized, as may be expected for systems in

26

Figure 22: Distribution of the path lines for Ra =103,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 23: Distribution of the path lines for Ra =103,ε=0.6 and 0.1 ≤R/L≤0.4.

which convection constitutes the major heat transfer mechanism (see Figs. 26-27 and Figs.

34-35). In this range of Ra values, secondary convective cells were sometimes generated,

while the temperature distribution along the radial direction was non-linear with clearly

recognizable single or multiple thermal plumes rising up from the top of the cylinder.

In summary, secondary convective cells never appeared for Ra≤104and/or R/L≤0.1.

For R/L=0.2, secondary convective cells appeared only for the value of ε=0.6 and for the

two values of R a =105and Ra =106. For R/L=0.3, secondary convective cells appeared

for the two values of ε=0.005 and ε=0.6 and the two values of Ra =105and Ra =106.

For R/L=0.4, secondary convective cells appeared for the two values of εand only for

Ra =106. In addition, it was observed that the ows characterized by ε=0.6 may contain

more secondary convective cells than their counterparts characterized by ε=0.005 - a

trend that never occurs the other way around. It thus appears that for ow separation

to occur high Ra,R/Land εvalues are required simultaneously. It is also noteworthy

that an increasing number of secondary convective cells with an increasing value of the Ra

number followed the same trend as that observed in Rayleigh-Bénard convection with an

increasing aspect ratio, i.e., for high values of R/L, the cylinder curvature can be locally

neglected, and the ow in the top region of the cavity resembles the Rayleigh-Bénard

conguration.

27

Figure 24: Distribution of the path lines for Ra =104,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 25: Distribution of the path lines for Ra =104,ε=0.6 and 0.1 ≤R/L≤0.4.

Figure 26: Distribution of the path lines for Ra =105,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 27: Distribution of the path lines for Ra =105,ε=0.6 and 0.1 ≤R/L≤0.4.

28

Figure 28: Distribution of the path lines for Ra =106,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 29: Distribution of the path lines for Ra =106,ε=0.6 and 0.1 ≤R/L≤0.4.

Figure 30: Distribution of the path lines for Ra =103,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 31: Distribution of the path lines for Ra =103,ε=0.6 and 0.1 ≤R/L≤0.4.

29

Figure 32: Distribution of the path lines for Ra =104,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 33: Distribution of the path lines for Ra =104,ε=0.6 and 0.1 ≤R/L≤0.4.

Figure 34: Distribution of the path lines for Ra =105,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 35: Distribution of the path lines for Ra =105,ε=0.6 and 0.1 ≤R/L≤0.4.

30

Figure 36: Distribution of the path lines for Ra =106,ε=0.005 and 0.1 ≤R/L≤0.4.

Figure 37: Distribution of the path lines for Ra =106,ε=0.6 and 0.1 ≤R/L≤0.4.

5.2 Quantitative results and discussion

5.2.1 Calculation of the Nusselt number

In this section, the discussion is focused on the average Nucand Nuhvalues corresponding

to the Nusselt numbers calculated at the cold cavity walls and at the cylinder surface,

respectively. The calculation of Nucis based on the arithmetic average of the average

Nusselt numbers at every wall of the cavity, each obtained by the method presented in

subsection 4.2.2. The value of Nuhis obtained by taking into account the heat ux from

the cylinder’s surface. Therefore, the average hot Nusselt number, Nuhis:

Nuh=1

2εSSρCp

Td

L−TL

∆tdS.(5.1)

The procedure of calculation of the Nuhvalue does not require explicit calculation of the

temperature gradients at the surface of the hot cylinder, rather it is an integral part of

the IBM. Note that TLis the temperature obtained at a specic Lagrangian point by

interpolation of the corresponding predicted (i.e., obtained by not taking into account the

existence of the immersed body) Eulerian temperature, which enables the calculation of

Nuhfor both transient and steady-state ows.

5.2.2 Comparison of the results of the lowest-temperature-dierence cases and previous

studies

The results of the current study obtained for the value of ε=0.005 were compared with

those from corresponding studies of natural convection ow from a hot cylinder placed

31

Figure 38: Physical model of a hot cylinder inside a cold tube adapted from [28] and [43].

within a 3D cavity obtained by employing the Boussinesq approximation [28], [43] (see

Fig. 38). Interestingly, there is acceptable agreement between our 2D study and the 3D

results obtained in refs. [28], [43]in terms of Nuhand Nucvalues for the whole range of

Ra and R/L, as summarized in Tables 1-2. The maximal relative deviation between our

results and the results obtained in refs. [28], [43] was 19% and could be attributed to the

impact of the lateral walls in the 3D conguration suppressing the convective ow and

thus decreasing the total heat ux.

5.2.3 Analysis of the heat uxes in the ow domain

As mentioned in Section 2, , the walls of the cavity were maintained at a cold temperature

Tc=1−ε, while the walls of the cylinder that was placed in the center of the cavity were

maintained at a hot temperature Th=1+ε. Therefore, the direction of the heat ux was

from the cylinder surface toward the cavity surfaces. The value of heat ux at each cavity

surface will reect the characteristics of the specic ow regime and can thus be quantied

by the calculation of the average values of Nucnumbers for each wall. The Nucvalues

were calculated by the formulas given in subsection 4.2.2.

As may be expected from symmetry considerations, the Nu values obtained for the

left and right walls of the cavity were close to each other (see Tables 3-4) for the entire

range of Ra-R/Lvalues. At the same time, there were signicant dierences in the Nuc

values obtained for the bottom and top walls of the cavity (see Tables 5-6) for the entire

range of Ra-R/Lvalues, with the Nu values at the top always being higher than those

at the bottom. These dierences increased with increasing Ra values, and for Ra =106

could reach up to one order of magnitude. To summarize, as the convective heat transfer

became more pronounced with increasing Ra values, the top of the cavity started to play

a more dominant role in removing heat from the system, which was clearly reected in a

gradual increase in the corresponding Nu values.

32

Table 1: Comparison between the present and the previously published Nuhvalues averaged over the surface

of a hot cylinder placed within a cold cube for ε=0.005.

Non-Boussinesq Boussinesq Non-Boussinesq Boussinesq

R/L0.1 0.2

Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]

1046.4920 6.4880 6.2493 5.1990 5.1500 5.1184

10511.8700 11.6620 11.1380 7.7780 7.5800 7.2271

10618.1000 19.2500 18.3260 14.3500 13.3610 13.9370

R/L0.3 0.4

Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]

1046.2630 5.7304 5.8084 8.8840 8.5544 8.7030

1057.3740 6.5169 6.4790 9.1240 8.7643 8.7030

10613.3600 11.4010 11.2720 11.9200 10.8320 10.7160

Table 2: Comparison between the present and the previously published N ucvalues averaged over the surface

of a hot cylinder placed within a cold cube for ε=0.005.

Non-Boussinesq Boussinesq Non-Boussinesq Boussinesq

R/L0.1 0.2

Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]

1041.0345 1.0208 1.0201 1.6662 1.6188 1.6161

1051.9112 1.8360 1.8099 2.5649 2.3814 2.3766

1062.8683 3.0348 2.9945 4.6134 4.3677 4.3985

R/L0.3 0.4

Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]

1042.8655 2.9091 2.6216 5.4591 5.3928 5.1919

1053.3675 3.0702 2.9726 5.6192 5.5131 5.2651

1066.1671 5.3844 5.1956 7.2361 6.8313 6.6106

33

Table 3: Nucon the left wall.

ε0.005 0.2

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.9389 1.5939 2.8186 5.4620 0.9877 1.5958 2.7040 5.4354

1040.9287 1.6347 2.8549 5.4550 1.0373 1.6550 2.7334 5.4341

1051.4945 2.3019 3.1750 5.5250 1.7238 2.2965 3.0942 5.5877

1062.2876 4.5233 6.0957 7.5126 2.3220 3.7539 5.3523 7.3707

ε0.4 0.6

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.9671 1.5791 2.6765 5.3313 0.9276 1.5618 2.6305 5.1832

1041.0173 1.6197 2.6896 5.3356 0.9774 1.5729 2.6453 5.1905

1051.7228 2.3632 3.1533 5.5551 1.7032 2.3394 3.1851 5.4981

1062.3817 3.7652 6.3741 7.0895 2.3033 4.6269 6.1530 7.9808

Table 4: Nucon the right wall.

ε0.005 0.2

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.9392 1.5945 2.8197 5.4640 0.9840 1.5976 2.7048 5.4374

1040.9291 1.6354 2.8559 5.4570 1.0337 1.6583 2.7340 5.4361

1051.4950 2.3030 3.1760 5.5270 1.7275 2.2982 3.0925 5.5897

1062.8784 4.5249 6.0970 7.5153 2.3751 3.6198 5.5117 7.3734

ε0.4 0.6

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.9612 1.5818 2.6775 5.3333 0.9235 1.5625 2.6315 5.1851

1041.0119 1.6241 2.6905 5.3375 0.9750 1.0880 2.2705 5.1924

1051.7259 2.3645 3.1544 5.5571 1.7048 2.3403 3.1862 5.5001

1062.4127 3.8251 6.3767 7.0920 2.2621 4.6286 6.1556 7.9837

34

Table 5: Nucon the bottom wall.

ε0.005 0.2

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.8819 1.5548 2.7986 5.4534 0.9575 1.5546 2.6772 5.4229

1040.5861 1.3616 2.6723 5.3717 0.8554 1.3812 2.4982 5.3157

1050.2150 0.7787 1.9192 4.8322 0.2756 0.6655 1.7119 4.6725

1060.2358 0.7924 1.3864 7.5153 0.2299 0.3993 0.7655 3.1495

ε0.4 0.6

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1030.9308 1.5322 2.6421 5.3133 0.8861 1.4995 2.5864 5.1582

1040.8119 1.2999 2.3834 5.1648 0.7617 1.0880 2.2705 4.9589

1050.2707 0.6223 1.2821 4.2901 0.2845 0.3204 1.1078 3.9579

1060.2032 0.4767 0.6584 2.6102 0.1486 0.1582 0.4593 2.0442

Table 6: Nucon the top wall.

ε0.005 0.2

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1031.0040 1.6374 2.8412 5.4732 1.0177 1.6432 2.7330 5.4502

1041.6941 2.0329 3.0787 5.5527 1.2933 2.0418 3.0288 5.5709

1054.4403 1.8762 5.1999 6.5926 4.2639 4.9223 5.3219 6.9002

1065.4828 8.6132 11.0893 10.2210 7.2042 9.7895 9.3328 11.2149

ε0.4 0.6

R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Ra

1031.0012 1.6333 2.7134 5.3519 0.9687 1.6273 2.6768 5.2105

1041.3028 2.0593 3.0640 5.5315 1.2711 2.1806 3.0940 5.4540

1053.8549 4.5273 5.0862 7.0268 3.3704 3.9956 4.8861 6.9461

1066.9131 9.2124 10.5519 10.8403 6.3005 7.8221 9.9085 10.7687

35

5.2.4 Approximation to the Nu-Ra power law

The results of the average Nusselt number for each conguration can be approximated,

by employing the least squares technique, to obtain the Nu−Ra power law. Following the

dimensional analysis stemming from the boundary layer theory, the Nu−Ra relationship

obeys the following power law [44]:

Nu f=C(Gr·Pr)m=C(Ra)m,(5.2)

where Gr,Pr are the Grashof and Prandtl numbers and Ra=GrPr, and Cand mare specic

constants, whose values depend on the Ra number and on the geometric properties of the

system under consideration.

Figures 39-42 present the distribution of the Nu number as a function of the Ra

number for the entire range of: ε∈ {0.005,0.2,0.4,0.6},Ra ∈ {104,105,106}and R/L∈

{0.1,0.2, 0.3,0.4}. Note that the Nu values obtained for the lowest value of R a =103were

not taken into account because of the dominance of the conductive heat transfer mechanism

(see also Figs. 22-23 and Figs. 30-31).

All Nu values were approximated to the Nu-Ra power law as formulated by Eq. (5.2).

Generally, acceptable accuracy was obtained when approximating the Nu−Ra relationship

by the power law for the entire range of εand R/Lvalues. The results obtained for smaller

cylinders (R/L=0.1,0.2) exhibited more precise power law ts, for which the smallest value

of R2was equal to R2=0.975. Remarkably, for all congurations characterized by these

R/Lvalues, Nu was approximately proportional to Ra0.22 , which is very close to the results

reported with respect to laminar natural convection in spherical shells [45–47].

Congurations with larger cylinders (R/L=0.3,0.4) exhibited slightly less pronounced

power law ts for the Nu −Ra relationship characterized by smallest value of R2, i.e.,

R2=0.84. Note also that Nu was approximately proportional to Ra0.16 and to Ra0.06

for the R/L=0.3 and R/L=0.4 geometries, respectively. Such a considerable decrease

in the heat ux rate for geometries characterized by large cylinders compared to these

characterized by smaller cylinders can be attributed to the blocking eects of both the

cylinder and the cavity boundaries, which suppress the momentum of the convective ow

(see e.g. [28], [43]).

5.2.5 Multiple steady-state regimes

Further numerical analysis revealed that steady non-Boussinesq natural convection ows

can exhibit multiple steady-state regimes. In particular, two independent steady-state

branches were found for the values of ε=0.4,R/L={0.2, 0.4}and Ra =106as shown in

Figs. 43-46 The stability of the revealed regimes was veried by randomly perturbing all

the ow variables with values deviating by about 10% from the corresponding steady-state

values and verifying that the ow then converged to the previously observed steady state.

Remarkably, while the corresponding steady states diered in terms of the number and

the size of the convective cells hosted within the ow domain, they were characterized

by very close Nu values, averaged over the cylinder and the cavity boundaries. It can

36

Figure 39: Nusselt vs. Rayleigh numbers for R/L=0.1.

Figure 40: Nusselt vs. Rayleigh numbers for R/L=0.2.

37

Figure 41: Nusselt vs. Rayleigh numbers for R/L=0.3.

Figure 42: Nusselt vs. Rayleigh numbers for R/L=0.4.

38

Figure 43: Flow and temperature patterns corresponding to two dierent steady state branches obtained for

ε=0.4,Ra =106and R/L=0.2.

Figure 44: Flow and temperature patterns corresponding to two dierent steady state branches obtained for

ε=0.4,Ra =106and R/L=0.3.

thus be concluded that both steady-state regimes obtained for the same values of the

governing parameters and belonging to the dierent branches were still characterized by

the same averaged heat uxes at all the ow boundaries. In summary, acceptable agree-

ment was obtained between our results for the lowest temperature-dierence cases and

results in the literature that were computed by applying the Boussinesq approximation

for the entire range of operating conditions and ow characteristics; this agreement veries

the suitability of our numerical methodology with an incorporated IBM for simulation of

compressible natural convection conned ows with complex geometry. The results of the

high-temperature-gradient cases showed good agreement with the boundary layer theory.

In addition, multiple congurations of the steady-state ow were discovered.

6 Conclusions

In the present work, a pressure-based solver for the simulation of the thermal compress-

ible natural convection non-Boussinesq ow of an ideal gas was developed. This solver

utilizes a second-order backward scheme and standard second-order nite volume method

for temporal and spatial discretizations, respectively. The novel pressure-corrected direct

39

Figure 45: Flow and temperature patterns corresponding to two dierent steady state branches obtained for

ε=0.4,Ra =106and R/L=0.4.

Figure 46: Flow and temperature patterns corresponding to two dierent steady state branches obtained for

ε=0.6,Ra =106and R/L=0.2.

40

forcing IBM developed by Riahi et al. [33] was adopted and extended to enforce the kine-

matic constraints of no-slip and of a given temperature on the surface of the immersed

body. The algorithm does not rely on the low-Mach-number assumption and does not

employ the methodology of splitting the pressure into hydrodynamic and thermodynamic

terms. Instead, all the equations are solved in their original fully compressible formula-

tions. Finally, the viscous heating is neglected, as is commonly done when simulating

ows characterized by low values of the shear stress.

The developed methodology was extensively veried by comparison with corresponding

independently obtained numerical data available in the literature for incompressible ows

[42] and non-Boussinesq compressible ows [4]. The pressure-corrected direct forcing IBM

was implemented for simulation of the natural convection non-Boussinesq ow developing

within a cold square cavity with a centrally located hot cylinder over a broad range of

governing parameters. The results obtained were analyzed qualitatively and quantitatively.

First, the spatial distributions of the path lines and temperature elds were obtained.

Second, the values of Nusselt number on the hot cylinder and the cold cavity surfaces

were calculated. Third, the thermal uxes at all the domain boundaries were quantied

by calculating the values of the corresponding Nusselt numbers. Finally, multiple steady-

state solutions for several congurations were discovered.

Two dierent strategies, one based on an iterative solution and the other, on a direct

solution of the discretized governing equations were applied in the current study. The

iterative solution utilized the BiCGstab method [38], while the direct solution was based

on the TPF method proposed by Lynch et al. [39] and subsequently adapted to conned

natural convection ows by Vitoshkin & Gelfgat in [40]. The major challenge in applying

the two methods lay in treating the non-linear terms and the temperature-dependent

coecients of the Helmholtz-like dierential operator, which were obviously not constant.

It was found that in the absence of an immersed body both the iterative solver and the

direct solver yielded suciently accurate solutions for non-Boussinesq ows. However, in

the presence of an immersed body, the iterative solver based on the BiCGstab method was

less sensitive to the time step values and provided more accurate results than its direct

counterpart based on the TPF method.

We summarize by giving a list of challenges that have been left out the scope of

the current study and need to be addressed further on the way of developing a system-

atic pressure-based numerical framework for simulating non-Boussinesq natural convection

ows. The rst challenge is the development of the method for simulating the ow around

moving bodies. In this case, it is well known that the direct forcing method can lead to

spurious non-physical oscillations if explicitly included in the fractional step approach (see

e.g. [34]). The problem can be remedied by utilizing semi-implicit formulation of the im-

mersed boundary method, which imposes kinematic constraints of no-slip on the predicted

non-solenoidal velocity eld up to machine zero precession [48]. The second challenge is

an extension of the currently presented method to simulate fully 3D natural convection

ows. Based on our previous work, which utilized an explicit implementation of the direct

forcing method to simulate steady state Boussinesq natural convection ows [49] we do not

41

Figure 47: Relative spatial error of the developed algorithm calculated for the values of ε=0.6 and Ra=106for

the θ(•),u(), and v(N) elds: (a) l2norm; (b) l∞norm.

1.0E-03

1.0E-02

1.0E-01

5.0E-03 5.0E-02

Spatial error

Grid step,

D

x

(b)

1.0E-03

1.0E-02

1.0E-01

5.0E-03 5.0E-02

Spatial error

Grid step,

D

x

(a)

see any objective reasons that could prevent the direct adaption of the currently presented

method for simulation of non-Boussinessq natural convection steady 3D ows. Again, in

the case of the appearance of spurious oscillations in the ow elds when simulating un-

steady natural convection ows, it may be necessary to switch from a completely explicit

implementation of the direct forcing to its semi-implicit counterpart (see e.g. [28]). Fi-

nally, the developed method needs further verication to simulate the ow around bodies

characterized by non-uniform curvature. In this case, the grid resolution must be adjusted

to a kernel of the selected discrete Dirac delta function to ensure that the results provided

by the interpolation and regularization operators acting on the ow elds are suciently

accurate.

7 Acknowledgements

The authors would like to thank the Israel Ministry of Energy for its nancial support for

this work (grant 218-11-038).

A Appendix: Estimation of the method accuracy

To further evaluate the accuracy of the steady-state solutions obtained in this study, we

present a formal analysis of the spatial accuracy of the results obtained. The analysis

has been performed by calculating the values of Euclidian and innity norms of relative

errors ∥Sex −Sapr∥/∥Sex ∥, where Sex corresponds to the most precise solution obtained

on 200×200 grid and Sapr corresponds to a series of approximate solutions obtained on

50×50,100 ×100 and 1500 ×150 grids. The data calculated for temperature and two

velocity components obtained for the steady state ow in a deferentially heated cavity

for the values of ε=0.6 and Ra =106is presented in Fig. 47. It can be seen that there

is a power-law relationship between ∆xand the norm of the relative error value. In all

presented cases, the value of the exponent is close to 2, which conrms the second order

spatial accuracy of the developed method.

42

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