The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid

In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.


Introduction
In computational fluid dynamics, one need to compute the drag exerted on a body in flow field; in particular, optimal shape design has received considerable attention already, see Li and Huang [1], Li, Chen and Yu [2], and Li, Su, Huang [3]. It has become vast enough to branch into several disciplines on the theoretical side, many results deal with the existence of solutions to the problem or its relaxed form, on the practical side, topological shape ℑ optimization which solves numerically the relaxed problem or by local shape variation. In this case we have to compute the velocity gradient 1 : u u n ∂ = ∂ along the normal to the surface of the boundary and normal stress tensor n σ to the surface. All those computation have to do in the boundary layer. Therefore this leads to make very fine mash; for example, 80% nodes will be concentred in a neighborhood of the surface of the body.
In this paper a boundary layer equations for 1 0 , u p p ℑ = on the surface will be established using local semi-geodesic coordinate system based on the surface, provide the computational formula for the drag functional. In addition, a dimensional split method for three dimensional Navier-Stokes equations is established by applying several 2D-3C partial differential equations on the two dimensional manifolds to approximate 3D Navier-Stokes equation.
The content of the paper is organized as the followings. Section 2 establishes semi-geodesic coordinate system and related the Navier-Stokes equations; Section 3 assumes that the solutions of Navier-Stokes equations in the boundary layer can be made Taylor expansion with respect to transverse variable, derive the equations for the terms of Taylor expansion; Section 4 proves the existence of the solutions of the BLE; Section 5 provides the computational formula of the drag functional; Section 6 provide a dimensional splitting method for 3D Navier-Stokes equations; Section 7 provide some examples.

Navier-Stokes Equations and Its Variational Formulation in a Semi-Geodesic Coordinate System
Through this paper, we consider state steady incompressible Navier-Stokes equations and its variational formulation in a thin domain δ Ω , a strip with thickness δ and by a Lipchsitz continuous boundary (see ref. [1]) wherre , H K are mean curvature and Gaussian curvature of ℑ . Throughout this paper, we employ semi-geodesic coordinate system ( ) 3 , : based on the surface ℑ (see [1] and Figure 1 x α ξ is nonsingular.
In addition, we review the main notation. Greek indices and exponents belong to the set { } is noted a . Furthermore, the physical or geometric quantities with the asterisk * express the quantities on the manifold ℑ , for example, α * ∇ is covariant derivative on ℑ . Furthermore, the physical or geometric quantities with the asterisk * express the quantities on the manifold ℑ , for example, α * ∇ is covariant derivative on ℑ .
The following give the relations of differential operators in the space and on ℑ (see [1] ; .
The strain tensors of the vectors in ℜ and on ℑ can be expressed as In the semi-geodesic coordinate system (see next section), define the bilinear form ( ) , a ⋅ ⋅ and trilinear form or   ,  2  d d  2  d ,   ; ,  d ,   ,  ,  ,  , , , Then, the primitive variable variational formulation for Navier-Stokes Equations (2.1') is given by while the Navier-Stokes Equations (2.7) in semi-geodesic coordinate system are expressed as

Boundary Layer Equations
Assume that ℑ is a two dimensional manifold parameterized by . In the neighborhood of the orientate surface ℑ let define a surface ( ) δ ℑ : , is unit outward normal vector to It is obvious that ( ) δ ℑ ia a geodesic parallel surface of ℑ and the geodesic distance each other is equal to δ where δ is a small constant.In this paper we only consider exterior flow around a body occupied by Ω with a two dimensional manifold ℑ = ∂Ω without boundary. The boundary layer domain and five algebraic equations Associated variational formulations with (3.2) is given by where the bilinear forms defined by in semi-geodesic coordinate system based on ℑ and right term f can be made Taylor expansion with respect to the transverse variable ξ , , Theorem 2 Assume that the Assumption II is satisfied. Then six unknown of ( ) in (3.7) satisfy following system of the nonlinear partial differential equations which are called stream layer equations II (abbreviation SLE II) (interface equations): The right terms ( ) In particular, for flexible (slip condition 0 0 u ≠ ) boundary surface ℑ , neglect hight order terms and keep one order term of δ , then

) become
The Proof of Theorems 1 and 2 is neglected.

The Existence of the Solution
In this section we prove the existence of the weak solution of (3.2). To do that we consider variational formulation of (3.2). Let ( ) ( ) ( ) H D is a sobolev space of 1-order with periodic boundary condition. Since ( [14], Th.1.8.6) we claim ( ) Then corresponding variational formulation for (3.2) is given by , 0, 1, and the thickness δ of boundary domain small enough, then bilinear form Proof The proof of (4.4) can be found in ([1] [4]). It remain to prove (4.6). By virtue of the positive definition of metric tensor a αβ and assumption of lemma and using Hoelder inequality, we assert that Assume that the two-dimensional manifold ℑ is smooth enough such that the metric tensor a αβ of ℑ and curvature tensor b αβ satisfy . Then the bilinear forms ( ) ( ) then they are also coercive respectively where C is a constant independent of , U V having different meaning at different place and Proof Indeed it is enough to prove the coerciveness (4.8) since the continuous and symmetric are obvious by Hoelder inequality. Since Lemma 1, In view of Korn inequality on Riemann manifold (see [4] Th.1.7.9 ) To sum up, we conclude our proof. # Next we consider variational problem (4.2) corresponding to boundary layer Equation (3.4). Let The thickness δ of the boundary layer is small enough. Then bilinear form , : : and also satisfies following inequality where δ is small enough and parameters ( ) Proof It is easy to verify (4.12) by applying Hoelder inequality and Poincare inequality. It remains to prove (4.13). At the first, we recall that the assumptions of the lemma shows We assert that Using Young inequality By similar manner,  Cµβ α µβ δ λ α µλ α β It is easy to verify that (4.23) is satisfied if the parameters ( ) 0 0 , α β in the definition (4.1) are held Next we consider trilinear form. Taking into account of where δ is the thickness of boundary layer, 8 , , k C λ λ are constants defined in the followings.

Proof
We begin with constructing a sequence of approximate solutions by Galerkin's method. Since the space is a separable Hilbert space, there exist sequence ( ) such that: 1) for all 1 i ≥ , the elements 1 2 , , Φ Φ  are linearly independent; 2) the finite linear combinations of the are called a basis of the separable space. Denote by . Then we solve an approximate problem of (4.2) if δ is small enough. Then Then, the compactness of the embedding of ( ) Taking the limit of both sides of (4.30) implies Then U * is a solution of (4.2) and which satisfies is sequentially weakly continuous in ( ) V D can be found in [3].

Dimensional Split Method for Exterior Flow Problem around an Obstacle and a Two Scale Parallel Algorithms
In this section, we proposal a dimensional split algorithm for the three dimensional exterior flow around a obstacle occupied by 3 Ω ⊂ ℜ . Î = ∂Ω is a smooth surface of the obstacle and 3R = Ω ∪ Ω . Assume that Ω is decomposed by a series of geometric parallel surfaces , 1, 2, i i ℑ = into a series of stream layer The features of these systems are that the right terms of them depend upon the solution of next system, for example, the right term of kth-system depend upon the solution of ( ) 1 k + th. system. It is better to apply alterative iteration algorithm to solve these systems. (2) Solve system of ( ) , , F F F by using results obtained , then goto (2) to be continuous until reach certainly accuracy.

66
The variational formulations corresponding to (5.7) and (5.1) are given respectively by where the bilinear forms and linear form are defined by ; , , , There are two choices to do that (1) assume that Our aim is to give boundary conditions on N ℑ . Owing to (4.12) we claim On other hand, we show ( ) (2). Let assume that the flow outside of N ℑ is governed by Oseen equation where Φ is a fundamental solution of following equation where K α is a Bessel function of second kind. Then integral expressions of solutions of Oseen problem (5.9) are given by Here we employ Cartesian coordinate system is semi-geodesic coordinate. By the transformation of coordinate, , c ⋅ ⋅ defined by (5.23) is symmetric, continuous and coercive from ( ) ( ) 2  1  2  2  1  1  2  1 2  1  2  1  2  1  2  1 2,  1 2,   2  1 2  1  1  1  1  1 2,   , , Parallel algorithms. The Domain ia made partition by m interfaces surfaces and we obtain 1 m + the systems of BLE I and SLE II. Solving each BLE I and SLE II independently, then applying alternatively iterative algorithm are performance at the same time. On the other hand, the parallel algorithms for BLE I and SLE II can be used. Therefore, parallel algorithms are applied in two direction at the same time.

Computation of the Drag
The drag is a force exerted on a solid boundary surface, for example, 0 ℑ . There is normal stress on 0 ℑ which can expressed under semi-geodesic coordinate based on ℑ by

The Flow around a Sphere
Assume that ( ) , , x y z and ( ) The tensor of second fundamental form, i.e. curvature tensor of spherical surface is given by ( ) 2  22  11  12  11  22  12  3  3  3   1  2  1  2  2  2  1  2  2  1   2   1  1  , sin , 0, , , 0, sin 1 , 0, det sin , We remainder have to give the covariant derivatives of the velocity field, Laplace-Betrami operator and trace-Laplace operator. To do this we have to give the first and second kind of Christoffel symbols on the spherical surface ℑ as a two dimensional manifolds The associated Laplace-Betrami operator and divergence operator on ℑ are given by while trace-Laplace operator on ℑ is a two points boundary value problem for ordinary differential equations.

The Flow around an Ellipsoid
Let parametric equation of the ellipsoid be given by , The metric tensor of the ellipsoid is given by   That is   2  1  3 , ,

= =
Firstly the numerical solution of boundary layer equations is validated quantitatively by comparison with results in references and finite element method. Table 1 presents results of pressure and total drag coefficients for various Reynolds numbers at 0 0.5 ξ = . Table 2 presents results of pressure and total drag coefficients for various values of 0 ξ at Re 1.0 = . An excellent agreement between the present results and that of Alassar and Badr [13] are both achieved. And the normal stress tensor ( ) h δ ℑ to the supper surface of boundary layer is considered as the boundary condition of boundary layer equations, which is obtained from the solutions of finite element method. According to Table 1 and Table 2 the precision of drag computation with boundary layer equations is higher than the finite element method, so the boundary layer equations could be used to improve the computation precision of flow in the boundary layer with low cost. Figure 2 presents the nearly stationary streamline patterns and pressure distributions at different Reynolds numbers 10, 30, 60 and 100 respectively for 0 0.5 ξ = . Here we note that our streamline patterns are similar to those obtained by Rimon and Cheng [14] for the sphere, since the separation angles and wake lengths are in close agreement with each other. Figure 2(b) shows a clearly visible secondary vortex at Re 60 = , in this regard our result is also consistent with Rimon and Cheng's [14] in spite of the difference in the size of the wake. Furthermore, Figure 2(d) shows a nice structure which corresponds to the a phenomenon observed for the flow around a circular cylinder. Since secondary vortices appear only at relatively high Reynolds number, we may conclude that the wake region is much more active at higher Reynolds number rather than that the wake length has to increase with the Reynolds number.   Figure 5 shows shows the surface dimensionless pressure distributions for the case 0 0.5 ξ = when Re 10 = , 30, 60 and 100. As Re increases, the difference in the pressure between the front and the rear stagnation points increases. Figure 6 proposes the corresponding pressure distributions in 3D.   , 0.5, 1.0 and 1.5. The pressure distributions obtained by FEM and BLE are almost the same, however the absolute value of pressure in FEM is generally a little higher than these in BLE, which is consistent with the results in Table 2. Figure 9 proposes the corresponding pressure distributions in 3D.  Finally, it has to be emphasized that since flow axisymmetry is assumed in the present study, none of our results give any indication about symmetry-breaking in a real flow. The presented method are, however, not restricted to axi-symmetric flow, the BLE I aforementioned could be used to compute the non-axisymmetric flow.