Development of immersed boundary methods for isothermal and thermal flows 5

With the calculated wall temperature, the problem subjected to Neumann condition is converted to a problem subjected to Dirichlet condition where the explicit direct forcing method used

3

Parts of materials have been published in

[1] C Shu, W.W Ren, W.M Yang, Int J Numer Meth Heat Fluid Flow, 23 (2013) 124-142

validated by applying it to simulate forced convection over a stationary heated

circular cylinder and natural convection in a horizontal concentric annulus

between two circular cylinders The obtained numerical results show that the

proposed IBM solver is suitable for addressing thermal flows subjected to heat

flux boundary condition accurately and efficiently

5.1 Methodology

5.1.1 Governing equations

The same flow configuration Ω+Γ as the one in Fig 2.1 is considered

Assume that a thermal fluid is flowing inside it Rather than specifying with

given temperatures as in Section 4.1.1, the boundary Γ herein is releasing

prescribed heat flux Q B in its outward normal direction to the surrounding

fluid Nevertheless, by representing the heated boundary Γ as a set of heat

sources at each boundary segment (represented by Lagrangian point), it shares

the same set of governing equations (4.1) – (4.3) and velocity boundary

condition (2.3) in the framework of IBM, while the temperature boundary

condition, under the current circumstance, is a Neumann-type one

where n points to the outward normal direction of Γ The heat source term

q in the energy equation (4.3), as in the case of thermal flows subject to

specified temperature condition, is distributed from the boundary heat

source ( ( ), )Δ XQ s t through (4.5)

The fluid flow field, i.e velocity field, can be calculated following the procedures suggested in Chapter 2 or 3 In the present discussion, we exclusively focus on the temperature field and energy equation

5.1.2 Heat Flux Correction Procedure

Predictor-corrector algorithm is a wonderful technique It is extremely useful when dealing with IBM and almost all the existing IBMs rely on the Predictor-Corrector algorithm to fulfill their implementation Following a similar predictor-corrector step (4.6) – (4.7) as for the case of thermal flows with specified temperature condition, the energy equation (4.3) together with temperature boundary condition (5.1) can be successfully solved once the

volumetric heat source q is known Therefore, the primary issue for the

whole solution process is the evaluation of boundary heat source Δ XQ( ( ), )s t

at each Lagrangian point, from which the volumetric heat source q could

become available through (4.5) However, the boundary heat source determination is not an easy job Several models have been examined during our preliminary study, and finally, an efficient heat flux correction-based solver, as will be elaborated in details in the following, is found to be effective and accurate

Before we discuss the evaluation of Δ XQ( ( ), )s t , let us look at why q is

introduced in the energy equation Note that Eq (4.6) is the standard energy equation for temperature without any heat source If T given in Eq (4.6) *

satisfies the heat flux condition (5.1), q should be taken as zero From this

process, it is clear that the non-zero value of q is due to the fact that the heat

flux condition (5.1) is not satisfied by T Indeed, it is from their difference *

So, at first, we need to calculate k T*( i,t)

n

∂

−

∂ X at each Lagrangian point To

do this, we can use discrete delta function interpolation (assuming the same spatial discretization as in Section 2.4.2 is utilized) to provide

∂ X represent temperature derivatives with

respect to x and y at Lagrangian point Xi , while ( )t

are temperature derivatives with respect to x and y at Eulerian

point xj Note that the temperature derivatives at Eulerian points are obtained

by the second order central difference schemes Finally, the temperature derivative at the Lagrangian point is calculated by

normal direction nG The Neumann temperature condition (5.1) is also related

to nG For the application of IBM, the whole domain including interior and

exterior of the immersed object is used as the computational domain Thus, at

a boundary point, there are two normal directions One is to point to the flow

domain while the other is to direct into the inside of immersed object The

boundary heat flux due to difference of Q B(X ,i t) and k T*( i,t)

n

∂

−

∂ X in the

two normal directions will both affect the temperature field at surrounding

Eulerian points Therefore, when the difference of Q B(X ,i t) and

Note from Eq (4.5) that the volumetric heat source q at the Eulerian grid

point is evaluated from the boundary heat source ΔQ through Dirac delta

function interpolation, which can be expressed in the following discrete form

( j, ) ( i, ) h( j i) i ( 1, , ; 1, , )

i

q x t =∑ΔQ X t D x −X Δs i= " M j = " N

Substituting Eq (5.4) into Eq (5.5) leads to

With calculated q from Eq (5.6), the temperature correction can be computed

from Eq (4.9), and the corrected temperature field is obtained by Eq (4.7)

It should be noted that although Zhang et al (2008) has ever applied the concept of immersed boundary to thermal flows with Neumann conditions, they suggested to first define a layer of assistant points which are placed one-grid spacing away from the immersed boundary along its outward normal direction With the help of these assistant points, the normal derivative of temperature in the Neumann condition is approximated by the first-order one-sided finite difference scheme, from which the wall temperature can be computed With the calculated wall temperature, the problem subjected to Neumann condition is converted to a problem subjected to Dirichlet condition where the explicit direct forcing method used in the work of Zhang & Zheng (2007) for isothermal flows is applied to correct the predicted temperature field to the corrected one While many extra procedures and efforts are required in their work, it is obvious that our proposed method is more efficient and straightforward

5.1.3 Computational Sequence

The basic solution procedure of the proposed method can be outlined below: 1) Use the solution procedures described in Chapter 2 or 3 to compute the velocity field u

2) Solve Eq (4.8) to get the predicted temperature T *

3) Use Eqs (5.2)-(5.3) to calculate T*( i,t)

n

∂

∂ X ( i = 1 , " , M ) and then substitute it into Eq (5.4) to compute the boundary heat flux ( i, )

Δ X (i=1,",M)

4) Calculate the heat source q(xj,t) (j=1,",N) using Eq (5.5)

5) Correct the fluid temperature at Eulerian points using Eq (4.7) Until now, both the velocity field and temperature field have been updated to time level n+1

6) Repeat steps (1) to (5) until a desired solution is achieved

The present boundary condition-implemented IBM, using velocity correction and flux correction technique, will be validated in this section through its application it to simulate both forced convection (forced convection over a stationary isoflux circular cylinder) and natural convection (natural convection

in a horizontal concentric or eccentric cylindrical annulus between an inner isoflux cylinder and an outer isothermal cylinder) problems Before we solve

convection problems, we will use a model heat conduction problem to investigate the spatial accuracy of the present solver

5.2.1 Numerical analysis of spatial accuracy

The heat conduction problem in Section 4.3.1 is once again used as a model example to investigate the spatial accuracy of proposed thermal IBM solver, where the governing equation is described by Eq (4.32) and the temperature

boundary conditions are specified as: T in 1

T = + x +y on the outer boundary The problem is solved using

five different uniform meshes with mesh spacing of 1 1 1 1

of 2, implying the second order of spatial accuracy

5.2.2 Forced convection over a stationary isoflux circular cylinder

Forced convective heat transfer from a stationary heated circular cylinder which is immersed in a cold free stream and releases constant and uniform heat flux is simulated for several low and moderate Reynolds numbers of

10

Re= 20, 40 and 100 and fixed Prandtl number of Pr =0.7 The setups in

the present problem are exactly the same as those in subsection 4.2.1 except

the boundary condition is replaced by the specified uniform heat flux Heat

transfer characteristics of isotherms, local Nusselt number distribution and

average Nusselt number on the cylinder surface are presented

In the simulation, the temperature is normalized by

B

T T T

Q D k

∞

−

where T∞ is the free stream temperature, and Q B is the specified uniform

heat flux at the cylinder surface in its normal direction The thermal condition

on the immersed boundary can then be expressed in the dimensionless form

X is the local convective coefficient Their specific

dimensionless forms are

Fig 5.2 shows isotherms in the vicinity of the cylinder for each case As can

be observed from Fig 5.2, the isotherms slightly cluster in the front surface of the cylinder, indicating a larger temperature gradient, or a higher heat transfer rate there than other regions Furthermore, with an increase of Reynolds number Re, the temperature around the cylinder surface is decreased From

Eq (5.12), we can say that the heat transfer is enhanced

Table 5.1 lists a comparison of computed average Nusselt numbers for Re =

10, 20, 40 with reference data in the literature (Ahmad & Qureshi 1992; Dhiman et al 2006; Bharti et al 2007) Fig 5.3 draws the local Nusselt number distribution on the cylinder surface along with the result of Bharti et al (2007) for Re = 10 and 20 All these results show a good agreement Fig 5.4 plots the time evolution of average Nusselt number for Re=100, which, once again, implies an obvious periodic variation of the flow field As

expected, the average Nusselt number Nu on the cylinder surface increases with Reynolds number Re

5.2.3 Natural convection in a concentric horizontal cylindrical annulus between an outer isothermal cylinder and an inner isoflux cylinder

The capability of present method is now tested by a natural convection problem Generally, natural convection is more complex than forced convection since its velocity and temperature fields are strongly coupled The buoyancy force is the driving force for the flow, and the Boussinesq approximation is often used Here, natural convection in a horizontal concentric cylindrical annulus is simulated The schematic view for the problem configuration is shown in Fig 5.5, where the surface of the inner cylinder with radius R i is maintained at a uniform heat flux Q B, and the outer cylinder of radius R o =2R i is kept at a constant temperature T∞ The flow behavior of this problem is characterized by Prandtl number Pr , Rayleigh

k

ρ βμ

= , where G=R o−R i is the gap width of the

annulus In this study, numerical investigations are carried out for three Rayleigh numbers of Ra=1000 ,5700 and 5× 104 while Prandtl number is kept at Pr=0.7 The initial conditions are set as zero for u and T for T ∞

in the whole computational domain The gap width G is taken as the reference

length, and the temperature is normalized by Eq (5.7) A uniform Eulerian

mesh with resolution h=G/ 64 and convergence criteria

As expected, the flow and thermal fields are symmetric about the vertical central line through the center of the annulus (Fig 5.6) A pair of crescent-shaped eddies are formed in the enclosure, one in each half When Rayleigh number is small (Ra=103), the heat flow in the enclosure is conduction-dominated, and the isotherms appear as a series of concentric circular-like shapes around the inner cylinder When Rayleigh number is increased (Ra=5700, 5 10× 4), buoyancy begins to play a more important role, and the thermal boundary layer on the bottom surface of the inner cylinder becomes thinner than that on its top surface Meanwhile, strong convection induces a plume on the upper part of the annulus Also, it is seen that with an increase of Rayleigh number, the plume becomes stronger and drives the flow impinging on the top wall of the outer cylinder, leading to a thinner thermal boundary layer and denser isotherm gradient around the surface of the inner cylinder and top wall of the outer one As a consequence, the heat transfer in these regions is enhanced These phenomena can be verified in Fig 5.7, which shows that maximum temperature on inner cylinder occurs at its uppermost point and the minimum temperature appears at its lowermost point for all the

three cases That is, at its lowermost point, the heat transfer rate is largest while at its uppermost point, the heat transfer rate is smallest Furthermore, Fig 5.7 reveals that the temperature at any location on the inner cylinder is always higher for larger Ra as compared to smaller one, indicating that the heat transfer rate increases with an increase of Ra

The local temperature distributions on the inner cylinder surface are displayed for Ra =5700,5×104 in Fig 5.8, while reference profiles in the literature (Yoo 2003) are also included Their comparison indicates that the results obtained by present method agree well with the reference data

5.2.4 Natural convection in an eccentric horizontal cylindrical annulus between an outer isothermal cylinder and an inner isoflux cylinder

In this subsection, the proposed method is further tested by another natural convection problem The geometry of the problem under consideration (as shown in Fig.5.9) is similar to the one investigated in Section 5.2.3, except that the two infinite horizontal cylinders are eccentrically arranged in vertical direction The eccentricity of the inner cylinder is denoted by e, whose positive value represents the upward direction The fluid flow and heat transfer

of the problem are characterized by the Rayleigh number Ra , Prandtl number Pr , radius ratio R R o/ i and eccentricity ε =e G/ , where Ra , Pr and G have the same definition as those in Section 5.2.3 In the present

study, numerical calculations are performed for four different Rayleigh numbers of Ra=103, 10 , 4 10 , 5 10 with the other three parameters fixed 6

at Pr=0.7, R o/R i =2.6 and ε = −0.625, corresponding to the numerical study of Ho et al (1989) Uniform Eulerian meshes with mesh size

/ 100

h=G are used for all the considered simulations, while the convergence criteria are set as &un+1−un &∞< ×1 10−5 and &T n+1−T n &∞< ×1 10−8

Fig 5.10 shows the streamlines and isotherms at various Rayleigh numbers It

is observed that for this eccentric geometry, the annular gap at the top region over the inner cylinder is enlarged, making the convective flow stronger there The qualitative features of isotherms illustrated above for the concentric geometry appear to be even more pronounced The local temperature profile along the inner cylinder surface is depicted in Fig 5.11 for different Ra Also included in the figure are the results of Ho et al (1989) for the purpose of comparison As expected, the present results are in good agreement with those

of Ho et al (1989) Finally, the average heat transfer rate over the inner cylinder surface is examined, which is represented by means of average Nusselt number defined as

B

in out

Q G Nu