An examination of natural convection within a spherical enclosure using hp-adaptive FEM

Xiuling Wang* and Darrell W. Pepper†

4505 Maryland Parkway, Box 454027, Las Vegas, NV 89154-4027

Numerical results are presented for natural convection within a spherical enclosure using an hp-adaptive finite element technique. The time dependent laminar 3-D equations of motion and energy for an incompressible fluid are solved. The hp-adaptive model is based on both mesh refinement and increasing spectral order to produce enhanced accuracy while attempting to minimize computational cost. The 3-D hp-adaptive FEM code is first benchmarked by solving the natural convection heat transfer problem in a differentially heated cavity. Results are in good agreement with the literature. Natural convection is simulated within a sphere heated from below and cooled from above. Four recirculation cells initially develop during the early phase of the transient solution.

Nomenclature B Body force h Characteristic of element length Fθ Load vector for temperature

vF Load vector for velocity

eh Element size

hk Diffusion coefficient

ek Streamline component of the diffusion tensor L Linear orthogonal projection operator M Mass matrix

iN Shape function p Shape function order P Pressure

rP Prandtl number

eP Peclet number r radial distance R Radius of a sphere

aR Rayleigh number t Time T Temperature V Velocity vector

*V Perturbed velocity vector x,y,z Cartesian coordinates α Petrov-Galerkin coefficient β Cell Peclet number

* Postdoctoral Fellow, NCACM, University of Nevada, Las Vegas 89154, AIAA Member † Professor and Director, NCACM, Department of Mechanical Engineering, University of Nevada, Las Vegas; AIAA Associate Fellow

American Institute of Aeronautics and Astronautics

1

9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-3783

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

ρ Density θ Non-dimensional temperature ∇ Divergence operator

1. Introduction atural convection heat transfer within spherical geometries is of practical interest in many engineering applications such as geophysics, nuclear engineering, chemical engineering and astrophysics. However, it is

difficult to ascertain flow patterns and temperature distributions within spherical enclosures, especially if the flow eventually bifurcates after some length of time [1, 2].

N In order to accurately simulate the flow and thermal patterns within a differentially heated sphere, an hp-adaptive finite element technology is employed based on both mesh refinement (h-adaptation) and increasing spectral order (p-adaptation). The 3-D hp-adaptive FEM algorithm is first validated by solving for natural convection within a differentially heated, rectangular cavity. Results show excellent agreement with data in the literature. Computational results are obtained and presented for flow and thermal patterns within a sphere heated along its bottom hemispherical surface and cooled on its upper hemispherical surface.

2. Governing Equations The following scaling relations are used to define the three-dimensional Navier-Stokes equations for laminar flow with heat transfer:

** * * *

2 2 2, , , ,/ / /

c

h c

T TX V p tX V p t TR R TR Rα ρα α T

−= = = = =

− (1)

where the Rayleigh number and Prandtl number are defined as:

( ) 3

,Prh cg T T RRa

β ναν α−

= = (2)

with R defined as the radius of the sphere, ν the kinematic viscosity,α the thermal diffusion coefficient, and β the thermal expansion coefficient. The non-dimensional governing equations are:

Continuity equation:

0V∇ ⋅ = (3) Momentum equation:

2Pr= −∇ + ∇ +iDV P VDt

B (4)

for natural convection the body force term in z direction is defined as: PrB Raθ≡ (5)

Energy equation:

2DDtθ θ= ∇ (6)

Equations (3) - (6) are solved within the spherical domain. The boundary conditions are specified as: 0V = at r R= (7)

where 2 2 2r x y z= + + and

at 0.50

0.5T

⎧⎪= ⎨⎪−⎩

, 0,, 0

r R zr R zr R z

0= <⎧

⎪ = =⎨⎪ = >⎩

(8)

The initial conditions are 0V T= = at 0t = (9)

American Institute of Aeronautics and Astronautics

2

3. Finite Element Model Hexahedral elements are used to discretize the problem domain. Replacing the variables and V θ with the trial

functions: i iV( , t) N ( )V (t)= ∑x x (10)

i i( , t) N ( ) (t)θ = θ∑x x (11) the weighted residual form for the momentum equation under projection decomposition (next section) is:

( )

Pr

Pr Pr 0

j jii j i i k k i i

j i j

ji i i i i

j

N NNN N d V N N V d V d V

x x x

NRa T N d N V n d

x

Ω Ω Ω

Ω Ω

⎛ ⎞ ⎛∂ ∂⎛ ⎞ ∂⎧ ⎫Ω + Ω + Ω⎜ ⎟ ⎜⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜∂ ∂ ∂⎩ ⎭⎝ ⎠ ⎝ ⎠ ⎝∂

− Ω− Γ =∂

∫ ∫ ∫

∫ ∫

i ⎞⎟⎟⎠ (12)

The weighted residual form for the energy equation is:

( )

0

j jii j i i k k i i

j i j

i

N NNN N d T N N V d T d T

x x x

N qd

Ω Ω Ω

Ω

⎛ ⎞ ⎛∂ ∂⎛ ⎞ ∂⎧ ⎫Ω + Ω + Ω⎜ ⎟ ⎜⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜∂ ∂ ∂⎩ ⎭⎝ ⎠ ⎝ ⎠ ⎝

− Γ =

∫ ∫ ∫

∫

i ⎞⎟⎟⎠ (13)

The corresponding matrix equivalent forms of the governing equations can be written as:

[ ] [ ] ( )( ) VM V K A V V F+ + =⎡ ⎤⎣ ⎦i

V (14)

[ ] [ ] ( )( ) M K A V Fθ θθ + + θ =⎡ ⎤⎣ ⎦i

(15)

where the matrix expressions are defined as: [ ] i j kM N N N d

Ω

= Ω∫ (16)

[ ] Pr i iV

i j

N NK d

x xΩ

∂ ∂= Ω

∂ ∂∫ (17)

[ ] i i

i j

N NK d

x xθΩ

∂ ∂= Ω

∂ ∂∫ (18)

( ) ( ) ji k k

j

NA V N N V d

xΩ

∂= Ω

∂∫ (19)

Pr jv i i

j

VF N n d

xΓ

∂= Γ

∂∫ (20)

iF N qdθΓ

= Γ∫ (21)

A Petrov-Galerkin scheme is employed to weight the advection terms in the momentum and energy equations in order to avoid dispersion or added diffusion common to standard Galerkin formulation. The Petrov-Galerkin weighting function is given as:

2

ei i i

hW N V N

Vα

= + ⋅∇⎡ ⎤⎣ ⎦ (22)

1coth ,2

e

e

V hK

α β ββ

= − = (23)

where eK is the streamline component of the diffusion tensor. Petrov-Galerkin upwinding alters the advection matrix in equation (19) to the form:

American Institute of Aeronautics and Astronautics

3

( ) ( ) ji k k

j

NA V W N V d

xΩ

∂= Ω

∂∫ (24)

Mass lumping is used to obtain a fully explicit time scheme. The inverse of the mass matrix becomes:

1 1[ ]i

Mm

− = (25)

4. Flow solver A projection-step algorithm is used for the Navier-Stokes flow solver. This method is based on the Helmholtz-

Hodge Decomposition Theorem (see Chorin et al [3] for details). The theorem states that a vector field, V on domain, can be uniquely decomposed in the form Ω

V U P= +∇ (26) where U has zero divergence and is parallel to the boundary ∂Ω .

Applying this concept to the Navier-Stokes equations, a linear orthogonal projection operator L is applied to both sides of the equation, i.e.

( )( 2PrVL P L V V Vt

∂⎛ ⎞+∇ = − ⋅∇ + ∇ +⎜ ⎟∂⎝ ⎠)B (27)

where . Equation (27) becomes: ( ) 0L P∇ =

( ) 2PrV V V Vt

∂ B= − ⋅∇ + ∇ +∂

(28)

Splitting the overall velocity into two components and , the momentum equation under the linear orthogonal projection operator L becomes:

*V V

1

2Prn n

n n nV V V V Vt

∗ + − B+ ⋅∇ = ∇ +∆

(29)

The projection of is a perturbed velocity onto the divergence free-space. Under the decomposition of the vector field

*V

( )*L V , one can make the projection

*V V t P= + ∆ ∇ (30) Taking the gradients of both sides of Equation (30), and since 0V∇ ⋅ = , a Poisson equation for P is obtained:

2 * /P V t∇ = −∇ ⋅ ∆ (31) In a discretized finite element representation, this can be written as:

*( )M V V P 0− +∇ = (32) where M is the mass matrix. Equation (32) can be rewritten as:

*( )M V V CPt

0− + =∆

(33)

where is the gradient operator. The equation is subject to the constraint of continuity C 0TC V = (34)

Applying mass lumping, (35) 1TC M CP C V− = *T

eventually permits the velocity to be solved using the relation (36) * 1V V tM CP−= − ∆

5. Adaptive Finite Element Technology There are four types of adaptation: (1) h-adaptation, where the element sizes vary while the order of the shape

functions are constant; (2) p-adaptation, where the element sizes are constant while the order of the shape functions increase to meet desired accuracy requirements; (3) r-adaptation, where the nodes are redistributed in an existing mesh in the process of adaptation; (4) hp-adaptation, which is the combination of both h-and p-adaptation. Numerous examples using h-adaptive FEM for solving CFD problems can be found in the literature [4] – [7].

American Institute of Aeronautics and Astronautics

4

How

ombined with p-adaptation: hp-adaptation. Specific adaptation rules must be followed in order to guarantee that the adaptive process can Some of the most important adaptation rules are briefly stated here.

I-irregular h-adaptive mesh rule is: an element can be refined only if its neighbors are at the same or higher level (I-irregular mesh); an element can bors are at the same or lower level (I-irregular mesh).

adaptation rule must be followed: the order for an edge common for two elements never exceeds orders of the neighboring middle nodes. For rectangular elem horizontal and vertical orders are considered.

tation rules for h- and p-adaptation are combined together in hp-adap

daptive mesh, and the final hp-adaptive mesh obtained by applying p-adaptive enrichments on the intermediate mesh. The p-adaptation is carried out when the solution is pre-asymptotic. The procedure is illustrated in Figure 1.

ever, limited work has been done using hp-adaptive FEM algorithms for solving momentum and energy transport problems [8] – [11].

Adaptation is a complex procedure, and becomes especially difficult when h-adaptation is c

be carried on successfully.

5.1. h-adaptation rules An unstructured anisotropic mesh is allowed which is an efficient, directional refined mesh where refinement in

one directional is needed. In h-adaptation, I-irregular mesh adaptation rule must be followed to avoid multiple constrained nodes (parent nodes themselves are constraint nodes also). The statement of

be unrefined only if its neigh

5.2. p-adaptation rules In p-adaptation hierarchical shape functions are used instead of the conventional shape functions commonly

employed in most finite element schemes. The employment of the hierarchical shape function allows the enrichment of the solution be obtained by adding new shape functions to the existing ones without changing the existing shape functions and DOF, but only add new ones. In p-adaptation, minimum mesh

ents in 2-D cases, both the

5.3. hp-adaptation rules As a combination of h- and p- adaptation, hp-adaptation can be either refined (unrefined) or enriched

(unenriched) whenever necessary. The adaptation. Constraints at the interface of elements supporting edge functions of different order are employed to

maintain continuity of global basis function. In this study, the hp-adaptation is carried on in three steps based on the temperature gradient. A sequence of

refinement steps is employed. Three consecutive hp-adaptive meshes are constructed for solving the system equations in order to reach a preset target error: initial coarse mesh, the intermediate h-a

American Institute of Aeronautics and Astronautics

5

Figure 1. Flow chart of hp-adaptation procedure

5.4. Adaptation strategy The hp-adaptation strategy used in this paper is an extension from the “three-step hp-adaptive strategy”

developed by Oden et al [8] – [11], in which the error estimator is based on the element residual method. In this study, an alternative “ ” norm error estimator is used to guide the adaptation procedure. 2L

An acceptable solution is reached when global and local error conditions are met [12]. A global error condition states that, global percentage error should not be greater than a maximum specified percentage error: maxη η≤ . If maxη η> , a new iteration is performed. The local error condition states that local relative percentage error of any

single element i

eσ should not be greater than the averaged error avge among all the elements in the domain. The average element error is defined as:

( ) 1/ 22 2*

maxavg

ee

m

σση

⎡ ⎤+⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(37)

A local element refinement indicator is defined to decide if a local refinement for an element is needed:

ii

avg

ee

ξ = (38)

when 1iξ > , the element is refined; when 1iξ < the element is unrefined. In an h-adaptive process, the new element size is calculated using:

1/old

new pi

hh

ξ= (39)

American Institute of Aeronautics and Astronautics

6

In a p-adaptive process, the new shape function order is calculated by: 1/ p

new old ip p ξ= (40)

6. Natural Convection Heat Transfer in a Cavity In order to demonstrate the accuracy of the proposed numerical algorithm, the model is first validated by solving

a well-known benchmark heat transfer problem: natural convection within a differentially heated cavity. Results are in excellent agreement with the literature.

Three dimensional natural convection is a popular benchmark case since it is an easy extension from 2-D [13]. Early solutions for 3-D natural convection in an enclosure were obtained using the three-dimensional equivalent of the vorticity-stream function form of the equations. One of the earliest simulations was conducted by Mallinson and de Vahl Davis [14], followed by efforts from many other researchers over several decades [15]-[17].

For comparison purposes, the simulation condition is taken as Ra=105 and Pr=0.71. The geometry and boundary setting for 3-D natural convection in the enclosure is shown in Figure 2 with ( 0 1,0 1,0 1x y z≤ ≤ ≤ ≤ ≤ ≤ ); the initial computation is based on a coarse mesh containing 1000 elements with initial order 1.

Figure 2. Computational geometry and boundary condition

The intermediate h-adaptive mesh is shown in Figure 3 (a) which consists of 12,634 elements, 11,718 nodes and 11,718 dofs. The final hp-adaptive mesh is shown in Figure 3 (b) which consists of 12,634 elements, 16,193 nodes and 29,108 dofs. Notice that the mesh is refined and enriched along the walls of the cavity due to high temperature and velocity gradients.

(a) (b)

Figure 3. Adaptive meshes (a) intermediate h-adaptive mesh; (b) final hp-adaptive mesh

American Institute of Aeronautics and Astronautics

7

(a) (b) (c)

Figure 4. Simulation results in mid-plane on final mesh (a) isotherms (b) velocity vectors (c) isobars

(a) (b) (c)

Figure 5. Simulation results for y=-0.4, y=0.2 and z=0 on final mesh (a) isotherms (b) velocity vectors (c) isobars

Quantitative comparisons for velocities and Nusselt number are shown in Table 1.

Table 1. Comparison of simulation results with benchmark data Case Ra umax vmax Nuave

2-D [10] 105 35.0 68.5 4.55 3-D [12] 105 41.0 69.8 4.62

Present 3-D 105 37.5 69.2 4.58

7. Natural Convection Heat Transfer in a Sphere Simulation efforts for natural convection in a differentially heated sphere were subsequently undertaken. No experimental data is readily available for the current boundary settings. However, results are in good agreement with simulations obtained by Heinrich and Pepper [1]. In these simulations, Pr = 1 and . The computational mesh is shown in Figure 6 (a-c).

45 10Ra = ×

American Institute of Aeronautics and Astronautics

8

(a) (b) (c)

Figure 6. Initial computational meshes (a) initial mesh (b) intermediate h-adaptive mesh (c) final hp-adaptive mesh

A series of 2-D planar isotherm patterns and velocity vectors at -0.8, -0.6, -0.4, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8 along each major axis (x-y, x-z, y-z planes) are shown in Figures 7-10, where 1 1R− ≤ ≤ .

Figure 7. Isothermal patterns in x-y plane

American Institute of Aeronautics and Astronautics

9

Figure 8. Velocity vectors in x-y plane

Figure 9. Isothermals in y-z plane

American Institute of Aeronautics and Astronautics

10

Figure 10. Velocity vectors in y-z plane

Figure 11. Isothermals in x-z plane

American Institute of Aeronautics and Astronautics

11

Figure 12. Velocity vectors in x-z plane

Preliminary results are obtained at t = 1.6 sec. Velocity vectors in x-z and x-y plane show the development of four recirculation cells within the spherical enclosure. Symmetrical characteristics can be observed in x-z and x-y planes for both the velocity vectors and isotherms, and agrees with previous results [1]. However, the employment of adaptive FEM significantly enhances the computational resolution. Further efforts will focus on simulations at later time periods and higher Ra values. It is anticipated that the flow and thermal patterns will bifurcate, creating an oscillatory motion. This was demonstrated for 2-D flow within a cylinder by Heinrich and Yu [17].

8. Conclusion In this study, a 3-D hp-adaptive finite element algorithm has been developed and applied to natural convection problems. The hp-adaptive algorithm refines and enriches meshes automatically based on key variable gradient and error distributions. The algorithm is first validated by solving for natural convection within a differentially heated cavity. The model is then used to simulate fluid motion and temperature patterns within a differentially heated sphere. Velocity vectors and isotherms are depicted at different locations of the main axis in the sphere and show the development of four recirculating cells early in the transient solution.

References 1 Heinrich, J. C. and Pepper, D. W. (1989), “Flow visualization of natural convection in a differentially heated

sphere”, FED-Vol.85, WAM ASME, San Francisco, California, Dec. 10-15. 2 Min, J. H and Kulacki, F. A. (1979), “Transient natural convection in a single-phase heat-generating pool

bounded from below by a segment of a sphere”, Nuc. Engr. Design, Vol. 54, pp.267-278. 3 Chorin, A. J. and Manderson, J. E. (1993), A Mathematical Introduction to Fluid Mechanics, 3rd edition,

Springer-Verlag, N. Y. pp.36-39. 4 Pelletier, D, Ilinca, F. and Hetu, JF. (1994), “Adaptive remeshing for convective heat transfer with variable

fluid properties”, AIAA J. Thermophy. and Heat Transfer, Vol. 8, No. 4, Oct. –Dec..

American Institute of Aeronautics and Astronautics

12

5 Pepper, D. W. and Stephenson, D. E. (1995), “An adaptive finite element model for calculating subsurface transport of contaminant”, Ground Water, Vol. 33, pp.486-496.

6 Pepper, D. W. and Carrington, D. B. (1999), “Application of h-adaptation for environmental fluid flow and species transport”, Int. J. Num. Meth. Fluids, Vol. 31, pp. 275-283.

7 Nithiarasu, P, and Zienkiewics, O. C., (2000) “Adaptive mesh generation for fluid mechanics problems”, Int. J. Num. Meth. Engr. Vol. 47, pp. 629 - 662.

8 Oden, J. T., Wu, W. and Ainsworth, W. (1995), Three-Step H-P Adaptive Strategy for the Incompressible Navier-Stokes Equations, Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, Springer-Verlag, pp. 347-366.

9 Devloo, P., Oden, J. T. and Pattani, P. (1988), “An h-p Adaptive Finite Element Method for the Numerical Simulation of Compressible Flow”, Comp. Meth. Appl. Mech. Engr., Vol. 70, pp.203-235.

10 Oden, J. T. and Demkowicz, L (1991), “h-p Adaptive Finite Element Methods in Computational Fluid Dynamics”, Comp. Meth. Appl. Mech. Engr., Vol. 89, n 1-3, pp. 11-40.

11 Oden, J. T., Kennon, S. R., Tworzydlo, W. W., Bass, J. M. and Berry, C. (1993), “Progress on Adaptive hp-Finite Element Methods for the Incompressible Navier-Stokes Equations”, Comp. Mech., Vol. 11, pp. 421-432.

12 Onate, E. and Bugeda, G. (1994), “Mesh Optimality Criteria for Adaptive Finite Element Computations”, The Mathematics of Finite Elements and Applications, Wiley, Chichester, England, pp. 121-135, Chapter 7.

13 De Vahl Davis, G., “Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution”, Int. J. Num. Meth. Fluids, Vol. 3, 249-264, 1983.

14 Mallinson, G. D. and de Vahl Davis, G. (1977) “Three-dimensional natural convection in a box: a numerical study”, J. Fluid Mechanics, Vol. 83, pp.1-31.

15 Pepper, D. W. (1987), “Modeling of three-dimensional natural convection with a time-split finite-element technique”, Num. Heat Transfer, Vol. 11, No.1, pp.31-55.

16 Shaw, H. (1987), “Laminar mixed convection heat transfer in three-dimensional horizontal channel with a heated bottom”, Num. Heat Transfer, Part A, Vol. 23, pp.445-461.

17 Fusegi, T.; Hyun, J. M.; Kuwahara, K.; Farouk, B.Fusegi, T., Hyun, and J. M Kuwahaora (1991) “Numerical study of three-dimensional natural convection in a differentially heated cubical enclosure source”, Int. J. Heat Mass Transfer, Vol. 34, No. 6, p 1543-1557.

18 Heinrich, J. C. and Yu, C.C., (1988), “Finite Element simulations of buoyancy-driven flows with emphasis on natural convection in a horizontal circular cylinder,” Comp. Meth. Appl. Mech. Engr., Vol. 69, 1988, pp. 1-27.

American Institute of Aeronautics and Astronautics

13