Development of immersed boundary methods for isothermal and thermal flows 4

  Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Chapter A Boundary Condition-Enforced Immersed Boundary Method for Heat Transfer Problems with Dirichlet Condition2 Heat transfer holds an important position in numerous engineering applications. Their numerical simulations may present challenges when complex geometries are involved. In this chapter, the IBM is extended to solve heat transfer problems by modeling the heated immersed boundary as a set of heat sources which is added to the energy equation as the source term. Based on a temperature correction technique, a boundary condition-enforced immersed boundary solver is developed for heat transfer problems with Dirichlet-type boundary condition, in which the heat source/sink term is considered as unknown and determined implicitly such that the energy equation and the corresponding thermal boundary condition can be accurately satisfied. Furthermore, the critical issue of how to effectively calculate the average Nusselt number has not been properly addressed when IBM is applied                                                               Parts of materials in this chapter have been published in W.W. Ren, C. Shu, J. Wu, W. M. Yang, Computers & Fluids, 57 (2012) 40-51. 100      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition to solve thermal problems. Another important contribution is that efficient ways to calculate the average Nusselt number in the framework of IBM are suggested. They are based on temperature correction at Eulerian points and heat source/sink at Lagrangian points, which completely avoid evaluation of temperature gradients at the boundary points. The present method and proposed ways to calculate the average Nusselt number are validated by their applications to simulate forced convection over a stationary isothermal circular cylinder and natural convection in a concentric annulus between a square outer cylinder and a circular inner cylinder. The obtained numerical results are very accurate and stable, and agree well with available data in the literature. 4.1 Methodology 4.1.1 Governing equations Assume that a cold fluid is flowing over a heated immersed body Γ inside the two-dimensional domain Ω shown in Fig. 2.1. By extending Peskin's original idea, the IBM can be used for solving the thermal flow problem, in which the heated immersed boundary Γ is modeled as a set of singular heat sources/sinks ΔQ at each boundary segment (represented by Lagrangian points). The heat source/sink at the Lagrangian point is then distributed into the surrounding fluid as a volumetric heat source/sink q through the Dirac delta function interpolation. As a result, the heat and mass transfer in the 101      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition framework of IBM inside the domain Ω are governed by ρ( ∂u + (ui∇)u) = −∇p + μ∇ 2u + f ∂t (4.1a) ρ( ∂u + (ui∇)u) = −∇p + μ∇2u + ρ g(1 − β (T − T∞ )) + f ∂t (4.1b) ∇ iu = ρcp ( (4.2) ∂T + (ui∇)T ) = k∇2T + q ∂t (4.3) with no-slip condition (2.3) and Dirichlet thermal condition (4.4) on Γ T (X(s, t ), t ) = TB (X(s, t ), t ) . (4.4) Here Eqs. (4.1a), (4.2), (4.3) describe forced convection problems while Eqs. (4.1b), (4.2), (4.3) describe natural convection problems. In Eq. (4.1b), the Boussinesq approximation has been used. T denotes the fluid temperature. β is the thermal expansion coefficient due to temperature difference. g is the gravitational acceleration directed downward. c p is the specific heat at constant pressure and k is the thermal conductivity at the reference temperature T = T∞ . The heat source/sink term q in the energy equation (4.3) is the heat density transferred to the fluid from the immersed boundary, which, as mentioned, can be written as q ( x, t ) = ∫ ΔQ ( X ( s , t ), t )δ ( x − X ( s , t )) ds (4.5) Γ Here, ΔQ( X( s, t ), t ) is the boundary heat source/sink. The solution procedure for velocity field u has been well described in 102      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Chapters and (for two-dimensional problems, Eqs. (4.1) – (4.2) can be transformed into the stream function-vorticity formulation if desired). In the following, we exclusively focus on the calculation of temperature field and a detailed description on how to derive an accurate temperature field will be elaborated. 4.1.2 Temperature correction procedure The boundary condition-enforced IBM for thermal flows developed in this chapter aims at suggesting an IBM solver which can accurately satisfy both the energy equation (4.3) and the Dirichlet temperature condition (4.4). By using the fractional step approach, it is found that the solution of equation (4.3) can be obtained by the Predictor-Corrector steps: T* −T n + (ui∇ )T ) = k ∇ 2T Δt (4.6) T n +1 − T * =q Δt (4.7) ρcp ( ρcp The Predictor step (4.6) solves the energy equation without considering the heat source/sink term q. The resultant solution is noted as predicted * temperature T (x, t ) , which can be calculated by solving the following discretized equation ρcp ρcp T* −T n k =− ((u n +1 i∇ )T * + (u n +1 i∇ )T n ) + (∇ 2T * + ∇ 2T n ) Δt 2 (4.8) 103      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition where the second-order Crank-Nicolson scheme is applied to the terms (ui∇)T and k∇2T . n +1 To obtain the desired temperature T (x) , the corrector step (4.7) is implemented. It can be observed clearly from Eq. (4.7) that adding a heat source/sink is equivalent to making a temperature correction ΔT ( x ) = q (x) Δt , ρcp (4.9) which is responsible for correcting the predicted temperature field to the desire one through T n+1 (x) = T * (x) + ΔT (x) (4.10) The heat source/sink q at the Eulerian grid point, as shown in Eq. (4.5), is distributed from the boundary heat source/sink ΔQ through Dirac delta function interpolation and can be expressed in the discrete form as (if the same spatial discretization scheme as in Section 2.4.2 is implemented) q ( x j ) = ∑ Δ Q ( X i ) D h ( x j − X i ) Δ si (i = 1, 2, , M ; j = 1, 2, ,N) i (4.11) Substituting Eq. (4.11) into Eq. (4.9) leads to ΔT (x j ) = ∑ i ΔQ ( X i ) Δt Dh ( x j − X i ) Δ si (i = 1, 2, ρcp , M ; j = 1, 2, ,N) (4.12) Note that the unknowns in Eq. (4.12) are the boundary heat sources/sinks ΔQ(Xi ) . Once they are determined, the temperature correction 104      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition and then corrected temperature can be calculated from Eqs. (4.12) and (4.10) respectively. So, the key issue that requires to be addressed in the proposed IBM model is the calculation of boundary heat sources. In an attempt to satisfy the temperature boundary condition (4.4), it should be ensured that the temperature at the boundary point interpolated from the corrected temperature field by the delta function interpolation is equal to the specified temperature TBn +1 ( Xi ) , that is, TBn +1 ( Xi ) = ∑ T n+1 (x j ) Dh (x j − Xi )h2 (i = 1, , M ; j = 1, , N) j (4.13) Substituting Eqs. (4.10) and (4.12) into Eq. (4.13) gives TBn +1 ( Xi ) = ∑ T * (x j ) Dh (x j − Xi )h j + ∑∑ j (i = 1, k ΔQ( X k )Δt Dh (x j − Xk )Δsk Dh (x j − Xi )h ρcp , M ; j = 1, (4.14) , N) Equation (4.14) can be put in the following matrix form [CT ][QT ] = [ DT ] (4.15) where ⎛ Dh11 ⎜ 21 Δt ⎜ Dh h [ CT ] = ρcp ⎜ ⎜⎜ M ⎝ Dh Dh12 Dh22 DhM Dh1 N ⎞ ⎛ Dh11Δs1 ⎟⎜ Dh2 N ⎟ ⎜ Dh21Δs1 ⎟⎜ ⎟⎜ DhMN ⎟⎠ ⎜⎝ DhN 1Δs1 Dh12 Δs2 Dh22 Δs2 DhN Δs2 Dh1M ΔsM ⎞ ⎟ Dh2 M ΔsM ⎟ ⎟ ⎟ DhNM ΔsM ⎟⎠ (4.16) 105      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition ⎛ TBn,1+1 ⎞ ⎛ Dh11 ⎜ n +1 ⎟ ⎜ TB ,2 ⎟ ⎜ Dh21 ⎜ [ DT ] = ⎜ ⎟ − h ⎜ ⎜⎜ n +1 ⎟⎟ ⎜⎜ M T B M , ⎝ Dh ⎝ ⎠ ⎛ ΔQ1 ⎜ ΔQ [QT ] = ⎜⎜ ⎜ ⎝ Δ QM TBn,+i (i = 1, Dh12 Dh22 DhM Dh1N ⎞ ⎛ T1* ⎞ ⎟⎜ ⎟ Dh2 N ⎟ ⎜ T2* ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ DhMN ⎟⎠ ⎜⎝ TN* ⎟⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ * , M ) , T j ( j = 1, (4.17) (4.18) , N ) and ΔQi (i = 1, , M ) represent TBn+1 (Xi ) , T * (x j ) and ΔQ(Xi ) , respectively. As shown above, the nature of temperature correction procedure is that the heat source/sink is determined in such a way that the temperature T (X(s, t ), t ) at the boundary point interpolated from the corrected temperature field T (x, t) equals to the specified boundary temperature TB (X(s, t ), t ) . In other words, the physical boundary condition is enforced. After the equation system (4.15) is solved, ΔQi at all Lagrangian points are obtained. They are then substituted into Eq. (4.12) to obtain the temperature correction ΔT j , which is further substituted into Eq. (4.10) to get the corrected temperature T j ( j = 1, , N). 4.1.3 Computational sequence The basic solution procedure of the proposed method can be outlined below: 1) Calculate the velocity field u using the method suggested in Chapter or 3; 106      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition 2) Solve Eq. (4.8) to get the predicted temperature T * . 3) Solve equation system (4.15) to obtain the boundary heat source ΔQi ( i = 1, , N ) at Lagrangian points and then use Eq. (4.12) to get the temperature correction ΔT . 4) Correct the fluid temperature at Eulerian points using Eq. (4.10). Until now, both the velocity field and temperature field have been updated to time level n + . 5) Use the corrected velocity and temperature as the initial conditions, and repeat steps to for the computation of next time level. The process continues until a converged solution is achieved (steady case) or the given time is reached (unsteady case). 4.2 Evaluation of Average Nusselt Number Nusselt number is a key parameter in the heat transfer problem. The local Nusselt number Nu is defined as Nu ( X( s, t ) ) = hc ( X( s, t ) ) Lref k (4.19) where hc ( X( s, t ) ) is the local convective heat transfer coefficient, Lref is the reference length. According to Newton's cooling law and Fourier's law, the heat conducted away from the wall by the fluid is equal to the heat convection from the wall, that is −k ∂T ( X(s, t ) ) = hc (X(s, t ))(TB − T∞ ) ∂n 107    (4.20)   Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Substituting Eq. (4.20) into Eq. (4.19) gives Nu ( X( s , t ) ) = TB − T∞ −k ∂T L ∂n ref = − Lref ∂T ( X( s , t ) ) k TB − T∞ ∂n The surface-averaged Nusselt number Nu (4.21) is an important parameter to examine the rate of heat transfer from the heated surface, which is defined as Nu = Ltotal ∂T ( X(s, t ) )ds B − T∞ ∂n ∫ Nu ( X(s, t ) ) ⋅ds = L ∫ − T Γ total Γ Lref (4.22) where Ltotal is the total length of the immersed boundary Ltotal = ∫ Γ (4.23) ds As shown in Eqs. (4.21) and (4.22), the evaluation of local and average Nusselt numbers involves the calculation of the temperature gradient at the boundary point. This chapter proposes two simple and efficient ways to calculate the average Nusselt number directly from the heat source at Eulerian points or the heat source at Lagrangian points, which completely avoid the evaluation of the temperature gradient at Lagrangian points. Numerical experiments show that the proposed ways can give very accurate and consistent results for the calculation of average Nusselt number. In the following, we will show the ways in detail. To simplify the illustration, the two-dimensional case is presented here. It should be noted that the suggested methods can be naturally extended to three-dimensional problems. 108      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition 4.2.1 Method 1: Direct evaluation of average Nusselt number from temperature correction at Eulerian points As shown in Eq. (4.5), the heat source at an Eulerian point is from the heat source at a few Lagrangian points. If we consider all the Eulerian points which receive heat from the immersed boundary, then from the energy conservation law, we have ∫ q ( x )dx = ∫ ΔQ ( X(s, t ) ) ds Ω (4.24) Γ in which the LHS (left hand side) is the volume integral in the whole fluid domain, and the RHS (right hand side) is the surface integral along the cylinder wall. According to Fourier's law, ΔQ ( X( s , t ) ) can be written as ΔQ ( X( s, t ) ) = −k ∂T ( X( s , t ) ) ∂n (4.25) Substituting Eq. (4.9) and Eq. (4.25) into Eq. (4.24) gives ∫ ρc Ω p ΔT ∂T dx = ∫ −k ( X( s, t ) )ds Δt ∂n Γ (4.26) The RHS of Eq. (4.26) can also be written in terms of local Nusselt number Nu ( X ( s , t ) ) as ∫ −k Γ Nu ( TB − T∞ )kLtotal k ⋅ Nu ( X( s , t ) ) (TB − T∞ ) ∂T ds = ( X( s, t ) )ds = ∫ , ∂n Lref Lref Γ (4.27) where TB is average temperature on the immersed boundary. So, Eq. (4.26) can be simplified to Nu ( TB − T∞ ) Lref kLtotal = ∫ ρ c p Ω ΔT dx Δt 109    (4.28)   Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition T n +1 − T n ∞ ≤ 1×10−8 . The spatial accuracy is measured by L1 norm of the relative error of temperature distribution, which is defined in the same way as in Griffith & Peskin (2005); Mori & Peskin (2008). Fig. 4.2 shows the relative L1 error of the numerical solution with respect to the mesh spacing. The slope of the line, which is 2, shows the second order of spatial accuracy. 4.3.2 Forced convection over a stationary isothermal circular cylinder Forced convection over a stationary circular cylinder has been extensively studied and used as a benchmark to examine the capability of new numerical methods. Many experimental (Roshko 1961; Grove & Shair 1964; Kuehn & Goldstein 1976; Sparrow et al. 2004) and numerical results (Dennis et al. 1968; Dennis & Chang 1970; Lange et al. 1998; Mittal 1999; Soares et al. 2005; Dhiman et al. 2006; Bharti et al. 2007) are available. This is a one-way interaction problem and only the velocity field can influence the temperature field. The fluid flow and heat transfer are characterized by the Reynolds number Re = ρU ∞ D μ and Prandtl number Pr = μc p k . The same computational domain as in Fig. 2.2 and the same domain discretization as in Section 2.6.1 is taken for numerical simulation. Meanwhile, the same velocity boundary conditions as those in Section 2.6.1 are applied. Generally, the temperature is normalized by T′ = T − T∞ TB − T∞ (4.33) 112      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition where TB is the uniform temperature on the cylinder surface and T∞ is the free stream temperature. Then the boundary conditions for temperature are set as: ∂T ′ =0 ∂y on the top and bottom boundaries T′ = at the inflow boundary ∂T ′ =0 ∂x at the outflow boundary TB′ = on the cylinder surface In this study, numerical simulations are conducted for several low Reynolds numbers of Re = 10, 20, 40 and a fixed Prandtl number of the convergence criteria T n +1 − T n ∞ ≤ 1×10−8 . Since Pr = 0.7 , with fluid flow characteristics like the streamlines, drag coefficient C D , recirculation length Lw / D behind the cylinder have already been reported in Section 2.6.1, here only heat flow characteristics such as isotherm patterns and average Nusselt number on the cylinder surface are presented and compared with published results (Dennis et al. 1968; Lange et al. 1998; Sparrow et al. 2004; Soares et a. 2005; Bharti et al. 2007). Fig. 4.3 shows isotherms in the vicinity of the cylinder at Re = 20, 40 . As can be seen, the temperature contours (isotherms) at different Reynolds number are topologically the same. They cluster heavily in the front surface of the cylinder, indicating a large temperature gradient there. This implies that the 113      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition heat transfer rate at the front surface of the cylinder is much larger than other regions. The clustering of isotherms is enhanced with increase of Reynolds number Re . As pointed out in Section 4.2, the average Nusselt number Nu is an important parameter for heat transfer problems. Two ways to evaluate the average Nusselt number are shown in Section 4.2. Their specific forms for this particular problem are Nu = RePr π ∑ ΔT j Δt j Δx j Δy j ( j = 1, (4.34) , N) and Nu = ∑ ΔQ Δ s i i π i (i = 1, (4.35) ,M) respectively. In the following, we will take the case of Re = 10 as an example to illustrate the performance of these two ways using different mesh sizes. The obtained numerical results using the two ways are shown in Table 4.1. It is obvious from Table 4.1 that Method and Method provide almost the same results. They converge well to stable solutions when the mesh size is refined. Table 4.2 lists the comparison of computed average Nusselt numbers for Re = 10, 20 and 40. Note that since the two methods give almost the same results, only the results obtained using Method are displayed. Table 4.2 shows that the present results basically agree well with reference data in the literature. As expected, the average Nusselt number 114    Nu on the cylinder   Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition surface increases with Reynolds number Re . 4.3.3 Natural convection in a concentric annulus between a square outer cylinder and a circular inner cylinder To further validate the capability of the proposed method, a natural convection in a concentric annulus between a square outer cylinder and a heated circular inner cylinder is simulated. Unlike the forced convection problem discussed in Section 4.3.2, the velocity and temperature fields are strongly coupled for the problem considered in this section. The buoyancy force is the driving force for the flow, and the Boussinesq approximation is used. The schematic diagram for the computational configuration is shown in Fig. 4.4. For this problem, heat is generated uniformly within the inner circular cylinder with radius r , which is placed concentrically within the cold square cylinder with side length L . The no-slip and isothermal conditions are imposed on both cylinder walls. The flow behaviour of this problem depends on the Prandtl number Pr , Rayleigh number Ra = c p ρ g β L3 (Tin − Tout ) kμ , and aspect ratio AR = L / (2r ) , where Tin is the temperature on the inner cylinder surface and Tout is the temperature on the outer cylinder surface, g is the amplitude of gravity acceleration. The aspect ratio AR can uniquely determine the configuration of the problem. In the present study, numerical investigations are conducted for three different aspect ratios ( AR = L / 2r = 5.0, 2.5, 1.67) and Rayleigh 115      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition numbers of Ra = 1×104 , 1×105 and 1×106 with Prandtl number fixed at Pr = 0.7 . The initial conditions are set to be zero for u and Tout for T in the entire domain. The side length of the outer cylinder, L, is taken as the reference length, and the temperature is normalized by T′ = T − Tout . Tin − Tout (4.36) Uniform meshes are used for all the simulated cases, with mesh sizes of h = L /128 for Ra = 1×104 and h = L / 256 for Ra = 1×105 and 1× 106 , respectively. The convergence criteria are set as T n +1 − T n ∞ u n +1 − u n ∞ < 1× 10−6 and < 1×10−8 . Numerical results in terms of streamlines, isotherms and average Nusselt number Nu are presented. For the concentric annulus, the flow and thermal fields are symmetric about the vertical central line through the center of the annulus. This can be observed clearly in Figs. 4.5-4.7, which show the streamlines and isotherms for different cases. When Ra = 10 as shown in Fig. 4.5, the heat transfer in the annulus is mainly dominated by conduction, and the velocity field is too weak to affect the temperature distribution. The isotherms in Fig. 4.5 show a series of concentric circular-like shapes around the inner cylinder. However, due to the effect of buoyancy, the thermal boundary layer on the bottom surface of the inner cylinder is slightly thinner than that on its top surface. The circulations of flow show that as the aspect 116      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition ratio Ar increases from 1.67 to , the distance between the inner cylinder and the walls of the outer cylinder increases. As a result, the two secondary eddies embedded in the primary eddy at AR = 1.67 and AR = 2.5 are merged into one primary eddy at AR = on both sides of the annulus. When Rayleigh number is increased to Ra = 105 as shown in Fig. 4.6, a plume begins to appear on top of the inner cylinder. As a result, the thermal boundary layer on the surface of the inner cylinder becomes thinner as compared to that at Ra = 10 . Also, the centers of symmetric eddies move upward. For the case of AR = 1.67 , the inner circular cylinder is large and the gap between the inner and outer cylinders is small. Therefore, the convective flow induces the formation of two additional symmetric vortices on the top of the annulus. For the case of AR = 2.5 , the two secondary eddies embedded in the primary eddy on both sides of the annulus at Ra = 10 are merged into a primary vortex at Ra = 105 . As Rayleigh number increases further up to Ra = 106 (Fig. 4.7), due to increasing effect of convection, the core of the symmetric primary vortices keeps moving upward. The plume formed on top of the inner cylinder becomes stronger and drives the flow impinging on the top walls of the outer cylinder, which leads to a thinner thermal boundary layer and denser isotherm gradient on the surface of the inner cylinder and top wall of the enclosure. As a consequence, the heat transfer in these regions is enhanced. The size of the pair of symmetric vortices formed on the upper side of the annulus for the case of AR = 1.67 at Ra = 105 greatly increases while the 117      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition inner secondary eddies become squeezed and small. Also, small vortices appear in the vicinity of bottom wall of the annulus for the case of AR = 2.5 and AR = owing to separation of strong convection flow. In addition, the size of the tiny vortices is slightly larger for AR = than that for AR = 2.5 due to larger gap between the inner and outer cylinder surfaces. To further demonstrate the capability of Method and Method 2, the average Nusselt number is also computed for this problem and compared with reference data in the literature. Here, the average Nusselt number defined as hin S in Nu = Nu is , where Sin is taken as half of the circumferential k length of the inner cylinder surface and hin is the average heat transfer coefficient of the inner cylinder surface. In this natural convection case, Eqs. (4.29) and (4.31) can be simplified as Nu = Nu = ∑ ρc ΔT j Δx j Δy j Δt ( j = 1, 2k (Tin − Tout ) j p ∑ ΔQ Δs i i i 2k (Tin − Tout ) (i = 1, , N) (4.37) (4.38) ,M) Note that for this problem, the side length is the reference length, Tin − Tout is the reference temperature, ρ c p L2 / k is the reference time. So, Nu can be expressed in terms of the non-dimensional variables as Nu = ∑ j ΔT j Δt Δx j Δ y j ( j = 1, , N) 118    (4.39)   Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Nu = ∑ ΔQ Δ s i i i (i = 1, (4.40) ,M) The computed average Nusselt number Nu using the two methods are compared with those of Shu & Zhu (2002) and Moukalled & Acharya (1996) in Table 4.3. It can be seen from the table that both Method and Method give exactly the same values of calculated average Nusselt number, which agree very well with the two sets of reference data. Table 4.3 also reveals that greatly depends on Rayleigh number Ra the average Nusselt number Nu and aspect ratio AR . increases with the increase of Ra due to the Nu effect of buoyancy-induced convection, while it decreases with the increase of AR due to the effect of widened annulus gap space. 4.4 Conclusions In this chapter, a boundary condition-enforced immersed boundary method is developed for heat transfer problems. The effect of thermal boundaries on the flow and temperature fields is considered through the velocity correction and temperature correction. In particular, a heat source term, which is distributed from the boundary heat source via a discrete delta function, is introduced into the energy equation. From the viewpoint of fractional step approach, this term is equivalent to making a correction in the temperature field. The temperature correction is evaluated implicitly in such a way that the temperature at the immersed boundary interpolated from the corrected temperature field satisfies 119      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition the physical boundary condition. Another important contribution of this chapter is that it presents simple and efficient ways to compute the average Nusselt number by directly using the temperature correction at Eulerian points and the boundary heat source at Lagrangian points. Note that the temperature correction at Eulerian points and the boundary heat source at Lagrangian points are part of the numerical solution. The present solver has proven to be of second order spatial accuracy through a numerical analysis. The efficiency and capability of the present method and ways to calculate average Nusselt number are validated by applying them to simulate both forced convection and natural convection problems. Numerical results showed good agreement with available data in the literature. It is believed that the present method has a promising potential for solving heat and mass transfer problems with Dirichlet-type boundary conditions. 120      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Table 4.1 Comparison of average Nusselt number obtained by the two proposed methods (Re=10) Eulerian mesh size No. of Lagrangian 560 × 400 840 × 600 1120 × 800 40 60 60 90 80 100 Method 1.9163 1.9160 1.9154 1.9158 1.9150 1.9150 Method 1.9162 1.9160 1.9153 1.9158 1.9138 1.9150 points Table 4.2 Comparison of average Nusselt numbers References Re=10 Re=20 Re=40 Present 1.9150 2.5238 3.3519 Dennis et al. (1968) 1.8673 2.5216 3.4317 Lange et al. (1998) 1.8101 2.4087 3.2805 Soares et al. (2005) 1.8600 2.4300 3.2000 Sparrow et al. (2004) 1.6026 2.2051 3.0821 Bharti et al. (2007) 1.8623 2.4653 3.2825 121      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Table 4.3 Comparison of computed average Nusselt numbers Case Ra 1×104 1×105 1×106 References Present Ar Moukalled & Method Method Zhu (2002) Acharya (1996) 1.67 5.303 5.303 5.40 5.826 2.5 3.161 3.161 3.24 3.331 2.051 2.051 2.08 2.071 1.67 6.171 6.171 6.21 6.212 2.5 4.836 4.836 4.86 5.08 3.704 3.704 3.79 3.825 1.67 11.857 11.857 12.00 11.62 2.5 8.546 8.546 8.90 9.374 5.944 5.944 6.11 6.107 122    Shu &   Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Fig.4.1 Configuration for the model problem slope=1.92 Fig. 4.2 L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 123      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition ( a) Re = 20 (b) Re = 40 Fig. 4.3 Isotherms for flow over a heated stationary cylinder at Re = 20, 40 Fig. 4.4 Schematic view of natural convection in a concentric annulus 124      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition (a) AR = 1.67 (b) AR = 2.5 (c) AR = Fig. 4.5 Streamlines (left) and isotherms (right) for Ra = 1×104 125      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition (a) AR = 1.67 (b) AR = 2.5 (c) AR = Fig. 4.6 Streamlines (left) and isotherms (right) for Ra = 1×105 126      Chapter A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition (a) AR = 1.67 (b) AR = 2.5 (c) AR = Fig. 4.7 Streamlines (left) and isotherms (right) for Ra = 1×106 127    [...]... 5.303 5 .40 5.826 3.161 3.161 3. 24 3.331 2.051 2.051 2.08 2.071 1.67 6.171 6.171 6.21 6.212 2.5 4. 836 4. 836 4. 86 5.08 5 3.7 04 3.7 04 3.79 3.825 1.67 11.857 11.857 12.00 11.62 2.5 8. 546 8. 546 8.90 9.3 74 5 1×106 Acharya (1996) 5 1×105 Zhu (2002) 2.5 1×1 04 Method 2 5. 944 5. 944 6.11 6.107 122      Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Fig .4. 1 Configuration... size of the pair of symmetric vortices formed on the upper side of the annulus for the case of AR = 1.67 at Ra = 105 greatly increases while the 117      Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition inner secondary eddies become squeezed and small Also, small vortices appear in the vicinity of bottom wall of the annulus for the case of AR = 2.5 and. .. numbers of Ra = 1×1 04 , 1×105 and 1×106 with Prandtl number fixed at Pr = 0.7 The initial conditions are set to be zero for u and Tout for T in the entire domain The side length of the outer cylinder, L, is taken as the reference length, and the temperature is normalized by T′ = T − Tout Tin − Tout (4. 36) Uniform meshes are used for all the simulated cases, with mesh sizes of h = L /128 for Ra = 1×1 04 and. .. Fig .4. 1 Configuration for the model problem slope=1.92 Fig 4. 2 L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 123      Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition ( a) Re = 20 (b) Re = 40 Fig 4. 3 Isotherms for flow over a heated stationary cylinder at Re = 20, 40 Fig 4. 4 Schematic view of natural convection... 1 24     Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition (a) AR = 1.67 (b) AR = 2.5 (c) AR = 5 Fig 4. 5 Streamlines (left) and isotherms (right) for Ra = 1×1 04 125      Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition (a) AR = 1.67 (b) AR = 2.5 (c) AR = 5 Fig 4. 6 Streamlines (left) and isotherms (right) for. .. Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition Table 4. 1 Comparison of average Nusselt number obtained by the two proposed methods (Re=10) Eulerian mesh size No of Lagrangian 560 × 40 0 840 × 600 1120 × 800 40 60 60 90 80 100 Method 1 1.9163 1.9160 1.91 54 1.9158 1.9150 1.9150 Method 2 1.9162 1.9160 1.9153 1.9158 1.9138 1.9150 points Table 4. 2 Comparison of. .. Γ (4. 30) The discrete form of Eq (48 ) is Nu = ∑ ΔQ Δs i i i kLtotal ( TB − T∞ ) Lref (i = 1, ,M) (4. 31) The above equation is also based on dimensional variables Its specific form for non-dimensional variables will be provided in the numerical examples 4. 3 Numerical Examples The proposed boundary condition-enforced IBM, using the temperature 110      Chapter 4 A Boundary Condition-Enforced IBM for. ..  Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition With approximation of volume integral by the mean theorem, Eq (4. 28) can be finally written as ∑ ρc ΔT j Δx j Δy j Δt Nu = Lref ( j = 1, kLtotal ( TB − T∞ ) p j , N) (4. 29) Note that variables on the RHS of Eq (4. 29) are dimensional The actual form of Eq (4. 29) for non-dimensional variables... to Ra = 106 (Fig 4. 7), due to increasing effect of convection, the core of the symmetric primary vortices keeps moving upward The plume formed on top of the inner cylinder becomes stronger and drives the flow impinging on the top walls of the outer cylinder, which leads to a thinner thermal boundary layer and denser isotherm gradient on the surface of the inner cylinder and top wall of the enclosure... cylinder and a heated circular inner cylinder is simulated Unlike the forced convection problem discussed in Section 4. 3.2, the velocity and temperature fields are strongly coupled for the problem considered in this section The buoyancy force is the driving force for the flow, and the Boussinesq approximation is used The schematic diagram for the computational configuration is shown in Fig 4. 4 For this . thermal condition (4. 4) on Γ ( (,),) ( (,),) B T stt T stt=XX . (4. 4) Here Eqs. (4. 1a), (4. 2), (4. 3) describe forced convection problems while Eqs. (4. 1b), (4. 2), (4. 3) describe natural.  Chapter 4 A Boundary Condition-Enforced IBM for Heat Transfer Problems with Dirichlet Condition 100  Chapter 4 A Boundary Condition-Enforced Immersed Boundary Method for Heat Transfer. ds LLTTn ∞ ΓΓ ∂ =⋅=− −∂ ∫∫ XX (4. 22) where tota l L is the total length of the immersed boundary total L ds Γ = ∫ (4. 23) As shown in Eqs. (4. 21) and (4. 22), the evaluation of local and average Nusselt