ISSN 09655425, Computational Mathematics and Mathematical Physics, 2015, Vol. 55, No. 10, pp. 1727–1732. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.V. Bogovalov, I.V. Tronin, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 10, pp. 1756–1761.
Rhie–Chow Interpolation in Strong Centrifugal Fields S. V. Bogovalov and I. V. Tronin National Research Nuclear University “MEPhI,” Kashirskoe sh. 31, Moscow, 115409 Russia email:
[email protected],
[email protected] Received September 3, 2014; in final form, March 3, 2015
Abstract—Rhie–Chow interpolation formulas are derived from the Navier–Stokes and continuity equations. These formulas are generalized to gas dynamics in strong centrifugal fields (as high as 106 g) occurring in gas centrifuges. DOI: 10.1134/S0965542515100085 Keywords: Navier–Stokes equations, Rhie–Chow interpolation formula, strong centrifugal fields in gas centrifuges, difference method.
1. INTRODUCTION Gas centrifuges (GCs) are used industrially to separate uranium isotopes. Centrifugal accelerations in them reach 106 g. One of the most promising tools for obtaining information on gas dynamics in GCs is the numerical simulation of gas flows in these devices. Accordingly, numerical methods for gas dynamics in strong centrifugal fields are being developed worldwide. Such fields arise when gas flows are computed in a rotating frame of reference. The necessity of computations in such a frame is caused by the tasks aris ing in the design of GCs. A secret of centrifugal isotope separation is that its effectiveness can be substantially improved by using a slow axial rotation flow in the GC. The main task in all gasdynamic computations related to GC design is to determine parameters under which the rotation flow ensures the maximum isotope separability. The gas speed in this flow reaches several millimeters per second near the rotor wall. The speed of the rotor is about 600–700 m/s. The simplest way to identify a rotation flow with such velocities against the back ground of a gas rotating at a speed of 600–700 m/s is to use a rotating frame of reference, where the solid state velocity of the gas is zero. For this reason, starting from pioneering works [1, 2] up to recent publica tions [3, 4], rotating frames were used to compute all gas flows in GCs. Until recently, rotation flows have been computed in the axisymmetric approximation. Relying on stateoftheart technologies, we can switch to computations in fully 3D geometry. Such computations have to be performed on grids with arbitrarily shaped cells in order to precisely reproduce the GC geom etry. The control volume method is usually used to discretize fluid dynamics equations on arbitrary grids. Accordingly, we need to compute the mass, momentum, and energy fluxes through the surfaces of control volumes. A key point of any numerical scheme is the interpolation of gasdynamic variables from nodes to the surface of control volumes. Every interpolation method involves pressure gradients, which reach huge values. For example, the pressure in the Iguassu model centrifuge varies by five or six orders of magnitude when the radius varies by 1 cm. The direct use of wellknown interpolation methods gives rise to unphys ical mass fluxes. As a result, computations with the required accuracy become impossible. Accordingly, a major problem in the design of specialized numerical schemes for gas flow simulation in GCs is the devel opment of an interpolation scheme for computing mass fluxes on the boundaries of control volumes. Proposed in the early 1980s, a popular method for computing mass fluxes through control volume sur faces makes use of nodal values of variables (see [5]). At present, this interpolation method is widely used in the majority of CFD schemes. At the same time, it has a number of shortcomings that are still to be over come. A serious methodological shortcoming of the method is that it is usually applied to a system pro duced by discretizing the Navier–Stokes, continuity, and energy equations. As applied to standard fluid dynamic problems, this approach does not usually face difficulties, since it relies on wellknown schemes that produce acceptable results. However, in the case of nonstandard problems, which include gas dynam ics in GCs, methodological difficulties are transformed into major technical problems that are difficult to resolve without clarifying the foundations of the method. 1727
1728
BOGOVALOV, TRONIN
In this paper, the interpolation relations proposed by Rhie and Chow are derived, in fact, for the first time, from the underlying principles and are used to obtain new interpolation relations for gas dynamic simulation in superstrong centrifugal fields reaching 106 g. Below, we derive the basic interpolation relations without their verification, which, for some reasons, was published earlier in two works. Specifically, the verification results were presented for steady gas flows in GCs in [6] and for unsteady problems in [7]. 2. RHIE–CHOW INTERPOLATION SCHEME IN STRONG CENTRIFUGAL FIELDS The gas flow in a strong centrifugal field is governed by the Navier–Stokes equation written in a rotat ing frame of reference, ∂ρv i ∂ρv i v k 2 ∂p ∂τ + + 2ρε ijk Ω j v k = – + ik + ρΩ r i , ∂x k ∂x i ∂x k ∂t
(1)
by the continuity equation ∂ ρ ∂ρv k + = 0 ∂t ∂x k and by the gas energy density equation, which is omitted, since it is not used in what follows (see [8]). Here, ρ is the density of the gas, vi are the velocity components, p is the gas pressure, Ω is the angular velocity of the GC rotor, ri are the components of the position vector projected onto a plane perpendicular to the rotation axis, and τik is the viscous stress tensor. For convenience, we introduce some notation. The term 2ρεijkΩjvk is represented as γikρvk, where γik = εijkΩj. Applying the divergence operation to Eq. (1) yields the equation 2 ∂ρv ∂ρv i v k ∂ ∂τ ik 2 ∂p ∂ ∂ k + + γ ik ρv k = – 2 + + ρΩ r i . ∂t ∂x k ∂x i ∂x k ∂x i ∂ ∂ x ∂ x x i k i ∂x i
(2)
The time derivative in the first term of this equation is discretized according to the formula n–1
n
∂ ∂ρv 1 ⎛ ∂ρv k ∂ρv k ⎞ k = – , ∂t ∂x k Δt ⎝ ∂x k ∂r k ⎠ n
(3)
n–1
where ρv i and ρv i are the mass flux densities at time levels n and n – 1, respectively. Using the conti nuity equation and Eq. (3), we reduce Eq. (2) to n–1 2 ⎛ ∂2 p ∂τ ik ∂ ρv i v k⎞ 2 n ∂ ρ ∂ρv k ∂ ∂ + = Δt ⎜ 2 – – ρΩ r i + γ ik ( ρv k ) + ⎟ . ∂t ∂x k ∂x i ∂x i ∂x k ⎠ ⎝ ∂x i ∂x i ∂x k ∂x i
which has the form of a continuity equation. The mass flux density at time level n is given by n
n
( ρv k ) = ( ρv k )
n–1
n
n ∂( τ ik ) ∂( ρv i v k ) ⎞ n 2 n ∂p + Δt ⎛ – + ρ Ω r k + – γ ki ( ρv i ) – . ⎝ ∂x k ∂x i ∂x i ⎠
(4)
This expression can be interpreted as the mass flux density on the surface of a control volume at time level n. Actually, this is an equation for the mass flux density (ρvk)n at time level n that involves the mass flux density at the preceding time level and the nodal values of variables. In what follows, this expression is regarded as the mass flux density at integration (IP) points on the surface of a control volume. A similar equation for nodal mass flux densities can be derived directly from the Navier–Stokes equation. 3. RHIE–CHOW SIMPLIFICATIONS In the original Rhie–Chow method, the viscous stress tensor is used in the simplified form 2
∂ vk ∂τ . ik = μ 2 ∂x i ∂x i COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 55
No. 10
2015
RHIE–CHOW INTERPOLATION IN STRONG CENTRIFUGAL FIELDS
1729
The viscous force is then discretized in space as ⎛ ∂τ ik = μ ⎜ ∂x i ⎝
∑w
nb v k ( nb )
– v kc
∑w nb
nb
⎞
nb⎟ ,
⎠
where wnb are the weights of the velocity at the points adjacent (neighboring) to the central point c. Here, c is a node or an IP point on the surface of a control volume. The basic idea underlying the Rhie–Chow interpolation method is that the mass flux averaged over nodal points is used to estimate the convective term and the term containing
∑
nb
w nb v i ( nb ) at an IP point.
Let Eq. (4) be written for nodes. The mass flux density determined by the nodal values is given by n
〈 ( ρv k ) 〉 ip = 〈 ( ρv k ) ⎛ ⎛ + ⎜ 〈 μ⎜ ⎝ ⎝
∑ nb
n–1
∂p + 〈 ρΩ 2 r 〉 〉 ip – Δt ⎛ 〈 〉 k ip ⎝ ∂x k ip
⎞ 1 w nb v k ( nb )⎟ 〉 ip – 〈 ρ c v kc〉 ip 〈 ρ ⎠ n
∑ nb
(5)
⎞ ∂ρv i v k ⎞ ⎞ w nb〉 ip⎟ – ⎛ 〈 γ ki ρv i〉 ip – 〈 〉 . ⎝ ∂x i ip⎠ ⎠ ⎠
Here, we also used the approximate equality n 〈 ( ρ c v kc ) 1 pc
∑w
nb〉 ip
nb
n 1 = 〈 ( ρ c v kc ) 〉 ip 〈 ρc
∑w
nb〉 ip .
nb
The average 〈…〉 in these equations is found according to some procedure, which is of no matter. Subtracting Eq. (5) from Eq. (4) yields n
n–1
n
( ρv k ) ip – 〈 ( ρv k ) 〉 ip = ( ρv k ) ip – 〈 ( ρv k ) ⎛ n – μ ⎜ ( ρv k ) ip ⎝
∑ nb
2 2 ∂p ∂p 〉 ip + Δt ⎛ – ⎛ ⎛ ⎞ ip – 〈 〉 ip ⎞ + ( ρΩ r k ) ip – 〈 ρΩ r k〉 ip ⎝ ⎝ ⎝ ∂x k⎠ ∂x k ⎠
n–1
w nb n 1 – 〈 ( ρv k ) 〉 ip 〈 ρc ρ ip
∑ nb
(6)
⎞ ⎞ n n w nb )〉 ip⎟ – γ ki ( ( ρv i ) ip – 〈 ( ρv i ) 〉 ip ) ⎟ . ⎠ ⎠
According to the Rhie–Chow technique, in the subtraction procedure, we set ∂ρv i v k ∂ρv i v k⎞ 〈 〉 ip = ⎛ ⎝ ∂x i ∂x i ⎠ ip and ⎛ 〈 μ⎜ ⎝
∑ nb
⎞ ⎛ w nb v i ( nb )⎟ 〉 ip = μ ⎜ ⎠ ⎝
∑ nb
⎞ w nb v i ( nb )⎟ . ⎠ ip
Simplifying (6) with the use of the formula w nb
∑ ρ nb
ip
1 = 〈 ρc
∑w
nb )〉 ip .
nb
we can rewrite it as n
n–1
n
( ρv k ) ip – 〈 ( ρv k ) 〉 ip = ( ρv k ) ip – 〈 ( ρv k )
2 ∂p ∂p 〉 ip + Δt ⎛ – ⎛ ⎛ ⎞ ip – 〈 〉 ip + ( ρΩ r k ) ip ⎝ ⎝ ⎝ ∂x k⎠ ∂x k
n–1
(7) 2
– 〈 ρΩ r k〉 ip – μ
∑ nb
w nb n n n n ( ( ρv k ) ip – 〈 ( ρv k ) 〉 ip ) – γ ki ( ( ρv i ) ip – 〈 ( ρv i ) 〉 ip ) ⎞ . ⎠ ρ ip
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 55
No. 10
2015
1730
BOGOVALOV, TRONIN
4. MODIFIED RHIE–CHOW INTERPOLATION The next step is to express the mass flux density at the IP point in explicit form. Let n
n
n
Y k = ( ρv k ) ip – 〈 ( ρv k ) 〉 ip , and 2 ∂p F i = ⎛ – + ρΩ r i⎞ . ⎝ ∂x i ⎠ ip
Then Eq. (7) can be rewritten as n–1
n
Yi = Yi
⎛ n n + Δt ⎜ ( F i – 〈 F i〉 ) – μ ⎝
w nb
Y ∑ ρ nb
n i
ip
n⎞ – γ ik Y k⎟ . ⎠
or ⎛⎛ 1 ⎜ ⎜ + μ ⎝ ⎝ Δt
∑ nb
n–1 ⎞ n w nb⎞ Yi n n ⎟ δ ik + γ ik⎟ Y k = + ( F i – 〈 F i〉 ). ρ ip ⎠ Δt ⎠
The new matrix Aik is defined as A ik = δ ik + β ip γ ik , 1 where β ip = ⎛ + μ ⎝ Δt
w nb⎞ –1 . Then the equation becomes nb ρ ip ⎠
∑
n n n β n–1 A ik Y k = Y i + β ( F i – 〈 F i〉 ). Δt
It is easy to see that the inverse of the matrix Aik is given by ( δ ik – β ip γ ik ) –1 . A ik = 2 2 ( 1 + β ip γ ik ) Then the solution of the equation is n –1 β n–1 –1 n n Y i = A ik ip Y k + A ik β ip ( F k – 〈 F k〉 ). Δt
(8)
It is convenient to represent this solution in another form. Since δik = Aik – βγik, the first term on the righthand side of the equation becomes β n–1 β n–1 –1 β –1 n–1 A ik ip Y k = ip Y i – A ik β ip γ km ip Y m . Δt Δt Δt
(9)
By the definition of β, we have β ip = 1 – β ip μ Δt
w nb
, ∑ ρ nb
ip
Substituting this relation into the last term of Eq. (9) gives ⎛ β –1 β n–1 –1 A ik ip Y k = ⎜ δ ik ip + A im β ip γ mk μ Δt Δt ⎝
w nb⎞
⎟ Y ∑ ρ ⎠ nb
n–1 k
–1
n–1
– A ik β ip γ km Y m .
ip
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 55
No. 10
2015
RHIE–CHOW INTERPOLATION IN STRONG CENTRIFUGAL FIELDS
1731
In turn, substituting this result into (8), we obtain ⎛ β n n –1 ρv i ( ip ) = 〈 ρv i 〉 ip + ⎜ δ ik + A im β ip γ mk μ ⎝ Δt –1
n–1
n
∑ nb
w nb⎞ n–1 n–1 ⎟ ( ρv k ( ip ) – 〈 ρv k 〉 ip ) ρ ip ⎠
n
(10)
n–1
+ A ik β ip ( F k ( ip ) – γ km ρv m ( ip ) – ( 〈 F k〉 ip – 〈 γ km ρv m 〉 ip ) ). Note that n
n n–1 n–1 n 2 δp F k ( ip ) – γ km ρv m ( ip ) = ⎛ – ∂ + δρ Ω r i⎞ + F k ( ip ) , ⎝ ∂x k ⎠ ip
(11)
where n–1
n–1 2 n–1 n–1 ∂p F k ( ip ) = – + ρ Ω r i, ip – γ km ρv m ( ip ) ∂x k ip
(12)
is the total force acting on the gas at the preceding time level, δpn = pn – pn – 1, and δρn = ρn – ρn – 1. With the help of formulas (10)–(12), the mass flux density at the IP point can be expressed in terms of the total force acting on the gas, the nodal mass flux densities, and the mass flux density at the IP point at the preceding time. Note that the resulting expressions differ widely from the standard Rhie–Chow interpolation formu las. In the case of no field, (10) becomes a scalar equation. It can be projected onto the normal to the con trol volume surface and the equations can be solved directly for the mass flux through this surface. In this case, there is no need to store all the components of the mass flux density. It is sufficient to store the total mass flux through each surface. In the case of high centrifugal accelerations, the situation differs substan tially. The mass flux density is expressed in terms of all its components at the preceding time level, so we have to store all the components of the mass flux density on each control volume surface. 5. CONCLUSIONS Based on the Rhie–Chow approach [5], the expression for the mass flux density on a control volume surface obtained in this work takes into account the contribution of not only the pressure gradient, as in the original method, but also of the centrifugal and Coriolis forces. A nontrivial fact is that the Coriolis force only at the preceding time level is involved in the mass flux density expression. At the nth time level, the mass flux density is determined only by the pressure gradient and the centrifugal force at this time level. Since the righthand side of the continuity equation involves the total force density, no unphysical mass flux densities arise on the control volume surfaces. The computational technique proposed has been tested on problems specially designed for testing numerical schemes for gas flow simulation in strong centrifugal fields. It has been found to perform well as applied to steady [6] and unsteady [7] flows. ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Science of the Russian Federation, task no. 3.726.2014/K. REFERENCES 1. T. Matsuda, T. Sakurai, and H. Takeda “Sourcesink flow in a gas centrifuge,” J. Fluid Mech. 69 (1), 197–208 (1975). 2. J. K. Park and J. M. Hyun, “Numerical solutions for thermally driven compressible flows in a rapidly rotating cylinder,” Fluid Dyn. Res. 6, 139–153 (1990). 3. S. Pradhan and V. Kumaran, “The generalized Onsager model for the secondary flow in a highspeed rotating cylinder,” J. Fluid Mech. 686, 109–159 (2011). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 55
No. 10
2015
1732
BOGOVALOV, TRONIN
4. W.C. Zhang, D.J. Jiang, S. Zeng, and Y. K. Jishu, “Numerical simulation of influence of feed position on flow field,” At. Energ. Sci. Technol. 48 (2), 331–335 (2014). 5. C. M. Rhie and W. L. Chow, “A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation,” AIAA. J. 21, 1525–1532 (1983). 6. S. V. Bogovalov, V. D. Borisevich, V. D. Borman, et al., “Verification of numerical codes for modeling of the flow and isotope separation in gas centrifuges,” Comput. Fluids 86, 177–184 (2013). 7. V. I. Abramov, S. V. Bogovalov, V. D. Borisevich, et al., “Verification of software codes for simulation of unsteady flows in a gas centrifuge,” Comput. Math. Math. Phys. 53 (6), 789–797 (2013). 8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; ButterworthHeinemann, Oxford, 1987).
Translated by I. Ruzanova
SPELL: OK
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
Vol. 55
No. 10
2015
