Development of immersed boundary methods for isothermal and thermal flows 5

  Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Chapter An Efficient Immersed Boundary Method for Thermal Flow Problems with Heat Flux Boundary Condition3 The exploration on generalizing Peskin’s original IBM to solve problems with Neumann-type boundary condition is discussed in this chapter and we propose a heat flux correction-based IBM for thermal flows with given heat flux boundary condition. By using the fractional step procedure, when the Neumann condition is not satisfied by the predicted temperature field, their difference will contribute as a boundary heat source. Then following the concept of conventional IBM, the boundary heat source will be distributed to the surrounding Eulerian points as a volumetric heat source, which will be used to correct the temperature field directly. In the whole process, the Neumann condition is treated directly within the scope of immersed boundary method. There is no need to define a layer of assistant points and to convert the Neumann condition to the wall temperature. The present method is                                                               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. [2] W.W. Ren, C. Shu, W.M. Yang, Int. J. Heat Mass Transfer, 64 (2013) 694-705.  128      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 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 QB 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 −k ∂T ( X( s, t )) = QB ( X( s, t )) on Γ , ∂n (5.1) 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 129      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition source ΔQ(X(s), t ) through (4.5). The fluid flow field, i.e. velocity field, can be calculated following the procedures suggested in Chapter 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 ΔQ(X(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. 130      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Before we discuss the evaluation of ΔQ(X(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 * ( Xi , t ) at each Lagrangian point. To ∂n this, we can use discrete delta function interpolation (assuming the same spatial discretization as in Section 2.4.2 is utilized) to provide ∂T * ∂T * ( Xi , t ) = ∑ ( x j , t ) Dh (x j − Xi )h2 (i = 1," , M ; j = 1," , N ) ∂x ∂x j (5.2a) ∂T * ∂T * t X , = ( i ) ∑ ( x j , t ) Dh (x j − Xi )h2 (i = 1," , M ; j = 1," , N ) ∂y ∂y j (5.2b) where * ∂T * ( Xi , t ) and ∂T ( X i , t ) represent temperature derivatives with ∂x ∂y ∂T * respect to x and y at Lagrangian point Xi , while (x j , t ) and ∂x ∂T * (x j , t ) are temperature derivatives with respect to x and y at Eulerian ∂y point x j . 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 131      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition ∂T * ∂T * ∂T * ( Xi , t ) = ( X i , t ) nxi + ( X i , t ) n yi ∂n ∂x ∂y Notice that in Eq. (5.3), the calculation of − k (5.3) ∂T * ( Xi , t ) depends on the ∂n 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 ∂T boundary heat flux due to difference of QB ( Xi , t ) and − k ( Xi , t ) in the ∂n * two normal directions will both affect the temperature field at surrounding Eulerian points. Therefore, when the difference of −k QB ( Xi , t ) and ∂T * ( Xi , t ) are considered in the two normal directions, ΔQ ( X i , t ) is ∂n calculated as ⎡ ⎛ ∂T * ⎞⎤ ⎡ ⎤ ∂T * ΔQ ( Xi , t ) = ⎢QB ( Xi , t ) − ⎜ −k ( Xi , t ) ⎟ ⎥ = ⎢QB ( Xi , t ) + k ( Xi , t )⎥ ∂n ∂n ⎝ ⎠⎦ ⎣ ⎦ ⎣ (5.4) 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 q ( x j , t ) = ∑ Δ Q ( X i , t ) D h ( x j − X i ) Δ si (i = 1, " , M ; j = 1, " , N ) i (5.5) 132      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Substituting Eq. (5.4) into Eq. (5.5) leads to ⎛ ⎞ ∂T * q (x j , t ) = ∑ ⎜ QB ( Xi , t ) + k ( Xi , t ) ⎟ Dh (x j − Xi )Δsi ∂n i ⎝ ⎠ (i = 1," , M ; j = 1," , N ) (5.6) 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. 133      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 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 or 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 * ( Xi , t ) ∂n (i =1,", M) and then substitute it into Eq. (5.4) to compute the boundary heat flux ΔQ ( X i , t ) (i = 1,", M ) . 4) Calculate the heat source q ( x j , 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 + . 6) Repeat steps (1) to (5) until a desired solution is achieved. 5.2 Numerical Examples 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 134      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 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: ( Tout = + ln x + y ∂Tin = on the inner boundary and ∂n ) on the outer boundary. The problem is solved using five different uniform meshes with mesh spacing of h = 1 1 , , , and 16 32 64 128 respectively. The spatial accuracy is measured by L1 norm of the 256 relative error, which is defined in the same way as in Griffith & Peskin (2005) and Mori & Peskin (2008). Fig. 5.1 shows the relative L1 error of the numerical solution with respect to the mesh spacing, which indicates a slope 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 135      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Re = 10, 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 T′ = T − T∞ QB D k (5.7) where T∞ is the free stream temperature, and QB 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 − ∂T ′ =1 ∂n (5.8) The local and average Nusselt numbers on the cylinder surface are defined as Nu ( X ( s ) ) = hc ( X ( s ) ) D (5.9) k and Nu = Nu ( X( s ) )ds 2π D ∫Γ where hc ( X ( s )) = (5.10) QB is the local convective coefficient. Their specific T − T∞ dimensionless forms are 136      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Nu ( X( s ) ) = Nu = 2π ′ T ( X( s ) ) 2π (5.11) ∫ T ′ (θ )dθ (5.12) that is, local Nusselt number is exactly the reciprocal of local dimensionless temperature. Therefore, for the problem of specified heat flux condition, the surface temperature is one of the important variables in the calculation, since it can reflect the heat transfer characteristics along the surface. 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 137      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition expected, the average Nusselt number on the cylinder surface Nu 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 Ri is maintained at a uniform heat flux QB , and the outer cylinder of radius Ro = 2Ri is kept at a constant temperature T∞ . The flow behavior of this problem is characterized by Prandtl number Pr , Rayleigh number Ra = c p ρ g β L3QB G / k kμ , where G = Ro − Ri is the gap width of the annulus. In this study, numerical investigations are carried out for three Rayleigh numbers of Ra = 1000, 5700 and 5×10 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 138      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition mesh with resolution h = G / 64 and convergence criteria & u n +1 − u n &∞ < 1× 10−6 and & T n +1 − T n &∞ < 1×10−8 is used for all the studied cases. Numerical results in terms of streamlines, isotherms and local temperature distribution on the inner cylinder are displayed. 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 ), 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 139      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 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 Ro / Ri and eccentricity ε = e / G , where Ra , Pr and G have the same definition as those in Section 5.2.3. In the present 140      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition study, numerical calculations are performed for four different Rayleigh numbers of Ra = 103 , 10 , 105 , 10 with the other three parameters fixed at Pr = 0.7 , Ro / Ri = 2.6 and ε = −0.625 , corresponding to the numerical study of Ho et al. (1989). Uniform Eulerian meshes with mesh size h = G /100 are used for all the considered simulations, while the convergence criteria are set as & u n +1 − u n &∞ < 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 Nu = QB G k ( Tin − Tout ) (5.13) The results obtained via Eq. (5.13) and corresponding data of Ho et al. (1989) are displayed in Table 5.2. Obviously, the present results match fairly well 141      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition with reference data of Ho et al. (1989). 5.3 Conclusions In this chapter, an efficient IBM is proposed for thermal flow problems with heat flux boundary condition. The effect of thermal boundaries to the flow and temperature fields is considered through velocity correction and heat flux correction in the framework of immersed boundary method. In particular, a heat source term, which is distributed from the offset heat flux at boundary via Dirac delta function interpolation, is introduced into the energy equation. The efficiency and capability of present method are validated by applying it to simulate both forced convection and natural convection problems. Heat flow characteristics in terms of local Nusselt number or temperature distribution are presented. The good agreement between present results and available data in the literature indicates that the present method provides a useful tool for solving heat transfer problems with Neumann boundary conditions. 142      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Table 5.1 Comparison of average Nusselt number Nu for Re = 10,20,40 Nu References Re = 10 Re = 20 Re = 40 Bharti et al. (2007) 2.0400 2.7788 3.7755 Ahmad et al. (1992) 2.0410 2.6620 3.4720 Dhiman et al. (2006) 2.1463 2.8630 3.7930 Present 2.0265 2.7413 3.7407 Table 5.2 Comparison of average Nusselt number Nu on the inner cylinder surface Nu References Ra = 103 Ra = 104 Ra = 105 Ra = 106 Present 2.3418 3.2052 4.5309 6.8879 Ho et al. (1989) 2.2724 3.2071 4.4821 6.8942 143      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition slope = 2.01 Fig. 5.1 the L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 144      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Re = 10 Re = 20 Re = 40 Re = 100 Fig. 5.2 Isotherms for flow over a heated stationary cylinder at different Re 145      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ θ Fig. 5.3 Comparison of local Nusselt number distribution on the cylinder surface for Re = 10, 20 146      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Fig. 5.4 Time evolution of average Nusselt number on cylinder surface for Re = 100 T∞ QB G Fig. 5.5 Schematic view of natural convection in a horizontal concentric cylindrical annulus 147      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Ra = 1000 Ra = 5700 Ra = 5× 10 Fig. 5.6 Streamlines (left) and isotherms (right) for different Ra 148      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ Fig. 5.7 Effect of Rayleigh number on local temperature distribution along the inner cylinder surface θ (a) Ra = 5700 149      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ (b) Ra = 50000 Fig. 5.8 Comparison of local temperature distribution on the inner cylinder surface for Ra = 5700 and 5×10 To Ro QB e Ri θ r Fig. 5.9 Configuration of natural convection in an eccentric horizontal cylindrical annulus 150      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Fig. 5.10 Streamlines (left) and isotherms (right) for different Ra 151      Chapter An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ Fig. 5.11 Comparison of temperature profile along the inner cylinder surface   152    [...]... surface for Re = 100 T∞ QB G Fig 5. 5 Schematic view of natural convection in a horizontal concentric cylindrical annulus 147      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Ra = 1000 Ra = 57 00 Ra = 5 10 4 Fig 5. 6 Streamlines (left) and isotherms (right) for different Ra 148      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary. .. horizontal cylindrical annulus 150       Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Fig 5. 10 Streamlines (left) and isotherms (right) for different Ra 151       Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ Fig 5. 11 Comparison of temperature profile along the inner cylinder surface   152     ... Condition θ Fig 5. 7 Effect of Rayleigh number on local temperature distribution along the inner cylinder surface θ (a) Ra = 57 00 149      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ (b) Ra = 50 000 Fig 5. 8 Comparison of local temperature distribution on the inner cylinder 4 surface for Ra = 57 00 and 5 10 To Ro QB e R i θ r Fig 5. 9 Configuration of natural convection... 100 Fig 5. 2 Isotherms for flow over a heated stationary cylinder at different Re 1 45     Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition θ θ Fig 5. 3 Comparison of local Nusselt number distribution on the cylinder surface for Re = 10, 20 146      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Fig 5. 4 Time evolution of average... ) (5. 13) The results obtained via Eq (5. 13) and corresponding data of Ho et al (1989) are displayed in Table 5. 2 Obviously, the present results match fairly well 141      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition with reference data of Ho et al (1989) 5. 3 Conclusions In this chapter, an efficient IBM is proposed for thermal flow problems with heat flux boundary. .. = 106 Present 2.3418 3.2 052 4 .53 09 6.8879 Ho et al (1989) 2.2724 3.2071 4.4821 6.8942 143      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition slope = 2.01 Fig 5. 1 the L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 144      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition Re = 10... 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 138      Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition mesh with u n +1 − u n ∞ resolution < 1× 10−6 and h = G / 64 T n +1 − T n ∞ and convergence criteria < 1×10−8 is used for all the... a uniform heat flux QB , and the outer cylinder of radius Ro = 2Ri is kept at a constant temperature T∞ The flow behavior of this problem is characterized by Prandtl number Pr , Rayleigh number Ra = c p ρ 2 g β L3QB G / k kμ , where G = Ro − Ri is the gap width of the annulus In this study, numerical investigations are carried out for three Rayleigh numbers of Ra = 1000, 57 00 and 5 10 while Prandtl... condition The effect of thermal boundaries to the flow and temperature fields is considered through velocity correction and heat flux correction in the framework of immersed boundary method In particular, a heat source term, which is distributed from the offset heat flux at boundary via Dirac delta function interpolation, is introduced into the energy equation The efficiency and capability of present method... 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 .  Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 128  Chapter 5 An Efficient Immersed Boundary Method for Thermal Flow Problems with Heat Flux Boundary. an inner isoflux cylinder and an outer isothermal cylinder) problems. Before we solve  Chapter 5 An Efficient IBM for Thermal Flow Problems with Heat Flux Boundary Condition 1 35  convection. 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  Chapter 5 An Efficient IBM for Thermal Flow