shock standoff distance increases with the magnetic field intensity, while the heat flux at the stagnation point decreases. Moreover, it is observed that the induced ...
Edge-based Finite Element Formulation of Hypersonic flows Under an Imposed Magnetic Field Wenbo Zhang1 and Wagdi G. Habashi2 NSERC-Lockheed Martin-Bell Helicopter Industrial Research Chair for Multi-physics Analysis and Design of Aerospace Systems CFD Laboratory, McGill University, Montreal, QC, H3A 2S6, Canada Nizar Ben Salah3 Laboratoire de Mécanique, Matériaux et Procédés, L’École Nationale Supérieure des Ingénieurs de Tunis, Université de Tunis, Tunis 1008, Tunisia
Dario Isola4 and Guido S. Baruzzi5 ANSYS, Inc., Montreal, QC, H3A 3G4, Canada Abstract The paper presents an edge-based Finite Element Method (FEM) for the simulation of three-dimensional hypersonic flows subject to an imposed magnetic field. Under the Magneto-Gas-Dynamics (MGD) assumptions, and at low-magnetic Reynolds numbers, a current continuity equation is employed instead of the system of Maxwell equations to obtain the induced electric field. The flow can be modeled by the Reynolds-Averaged Navier-Stokes (RANS) equations, and the electromagnetic body force and Joule heating resulting from the imposed magnetic field introduced through source terms. Both inviscid and viscous flows over a sphere, as well as an Apollo-like re-entry 1 1
PhD Candidate, Department of Mechanical Engineering, 688 Sherbrooke St. West.
2
Professor and Director CFD Laboratory, Chairholder, Department of Mechanical Engineering, 688 Sherbrooke St. West, AIAA Fellow.
3
Professor, Department of Mechanical Engineering.
4
Senior Software Developer, ANSYS Canada, 1000 Sherbrooke St. West.
5
Technical Fellow, ANSYS Canada, 1000 Sherbrooke St. West, AIAA Member.
capsule geometry, are used to validate the approach. For both geometries, inviscid and viscous results show that the shock standoff distance increases with the magnetic field intensity, while the heat flux at the stagnation point decreases. Moreover, it is observed that the induced electric field has a substantial impact on the effectiveness of MHD heat-shield and cannot be neglected in the calculations. An anisotropic mesh adaptive methodology is used to refine the results and demonstrate grid independence. 1.
Introduction Aviation is witnessing a resurgence of interest in devices that use magnetic and electric effects to control
hypersonic flight. The Thermal Protection System (TPS) of re-entry vehicles is perhaps the most important application [1-4], even though other designs are still used to protect the vehicle’s structure from thermal damage [5]. Extremely high temperatures trigger dissociation and ionization of the air as it traverses the bow shock in front of the vehicle. Although at weak levels of ionization the gas can still be considered as cold plasma, at hypersonic speeds the use of MGD-based devices constitutes an attractive alternative to improve aerodynamic and thermal performances. One such technique consists in a suitably applied magnetic field that induces an electromagnetic body force opposing the fluid motion across magnetic field lines, which in return reduces the drag and the peak heat transfer rate near the stagnation point. The general features of hypersonic flows are complex and cannot be addressed in terms of classical gas dynamics only, but require a multidisciplinary approach accounting for finite-rate chemistry, non-equilibrium thermodynamics, radiation and electro-magnetic effects [6-9]. Because it is rather challenging and expensive to reproduce these conditions in experimental facilities [10-11] Computational Fluid Dynamics (CFD) plays a significant role in understanding the complex physics of hypersonic flows and in designing high-speed vehicles [12-18]. The most general model that could be used in MGD simulations for aerospace applications makes use of the full Maxwell equations coupled with mass, momentum and energy conservation equations for an electrically conducting gas subject to an electromagnetic field. This model is referred to as the full magneto fluid dynamic (FMFD) model [19]. Under the MGD assumptions, the FMFD model can be simplified by assuming that displacement current, time variation of the charge density, and convection current are negligible. This results in the so-called simplified magneto fluid dynamic (SMFD) model. In this formulation, the magnetic induction equations replace the Maxwell equations to account for the convection and diffusion of the magnetic field. Dropping the displacement current from the FMFD
2
system effectively reduces the timescale of the electromagnetic problem to just the fluid timescale, making the resulting SMFD model significantly less stiff [20]. Solving the magnetic induction equations even in 2D cases is not a trivial task at low levels of electrical conductivity, typical of aerodynamic applications. Indeed, magnetic diffusion dominates the equation at low levels of ionization, since it is inversely proportional to electrical conductivity, and, if not carefully discretized, numerical instabilities may arise [21]. When simulating the effect of MGD devices for the design of a TPS, the low-magnetic Reynolds number (Rem) formulation [22] is often used, since the convection of the imposed magnetic field is negligible compared to its diffusion. The induced magnetic field is thus considered negligible and the current continuity equation is solved instead of the magnetic induction equation to obtain the induced electric field [4]. One can further assume that the induced electric field is also trivial and the total magnetic field can be replaced in Ohm’s law by the imposed one. Poggie and Gaitonde adopted this simplified version of low-magnetic Reynolds number approximation to investigate flows over a hemisphere [23]. An increase in shock standoff distance caused by the applied magnetic field was observed in both inviscid and viscous cases, as well as a reduction of the wall heat flux. Nevertheless, it has been suggested that the induced electric field must be calculated in a self-consistent manner utilizing the current continuity equation [2], but the impact of this induced field has yet been elucidated according to the authors’ best knowledge. Numerous attempts have been made in the development of codes for hypersonic flows within the FE framework. Ait-Ali-Yahia and Habashi [24] developed a fully implicit FE scheme adopting a two-temperature model and finiterate chemistry. Mesh adaptation techniques were used to finely resolve discontinuities in inviscid flow simulations, using structured grids.
Despite many advantages of the FE formulation (good representation of complex geometries, natural imposition of boundary conditions and second-order discretization of the viscous terms) significant difficulties remain in the design of shock-capturing second-order stabilization terms. Recent attempts were made by extending the SUPG formulation to include a discontinuity detector [25]. A different approach can be followed by resorting to the edgebased Galerkin formulation first developed by Selmin [26], which made use of upwinding schemes traditionally developed in the Finite Volume (FV) context. In this formulation mesh-dependent quantities are separated from the solution dependent ones, allowing the computation of the metric edge coefficients in the pre-processing phase. The
3
assembly is carried out by looping over edges instead of elements, considerably reducing the indirect addressing associated with unstructured meshes, and yielding substantial computational savings. In this paper, an edge-based FEM for the simulation of 3D hypersonic flows under an imposed magnetic field, at low magnetic Reynolds number, is presented. The effects of the induced electric field on MHD heat-shield are investigated. The paper is structured as follows. In section two, the mathematical modeling is presented with the electromagnetic forces introduced through source terms added to the RANS equations. The third section details the FE edge-based discretization and the solution strategy. In section four, three numerical tests are presented for inviscid and viscous flows over a sphere and an Apollo-like capsule re-entry. Conclusions are drawn in the last section. 2.
Mathematical Modeling
2.1. Flow field Hypersonic flows under an imposed magnetic field are modeled by the compressible Navier-Stokes equations and the magnetic induction equation. The former express the conservation of mass, momentum and energy of the fluid, while the latter is directly derived from the Maxwell equations under the MGD assumptions [27]. The complete set of these equations can be written as:
𝝏𝝆 + 𝜵 ⋅ (𝝆𝑽) = 𝟎 𝝏𝒕 𝝏𝝆𝑽 + 𝜵 ⋅ (𝝆𝑽𝑽 + 𝒑𝑰) − 𝜵 ⋅ 𝝉 = 𝒋 × 𝑩 𝝏𝒕 𝜕𝜌𝑒 t + 𝜵 ⋅ ((𝜌𝑒 𝑡 + 𝑝)𝑽) − 𝜵 ⋅ (𝑽 ⋅ 𝝉) + 𝜵 ⋅ 𝒒 = 𝒋 ⋅ 𝑬 𝜕𝑡
(1)
𝝏𝑩 − 𝜵 × (𝑽 × 𝑩) + 𝜼𝜵 × (𝜵 × 𝑩) = 𝟎 𝝏𝒕
where 𝜌 is the density, V the velocity field, 𝑒 𝑡 the total energy per unit mass, p the pressure, 𝝉 the stress tensor, q the heat flux, 𝜂 the magnetic diffusivity, 𝒋 the electric current density, 𝑩 the total magnetic field, equal to the imposed magnetic field 𝑩𝟎 plus the induced one 𝒃 and 𝑬 the electric field. The last equation of system (1) is the magnetic induction equation. It consists of a diffusive curl-curl term, a convective curl term and a time term, and it directly relates the hydrodynamic quantity 𝑽 to the electromagnetic one 𝑩. The
4
electromagnetic body force and Joule heating appearing on the right-hand side of the momentum equations, requires the computation of the electric current density which can be done according to Ohm’s law: 𝒋 = 𝝈 · (𝑬 + 𝑽 × 𝑩)
(2)
𝝈 being the tensor electrical conductivity and equal to a scalar if Hall effect is neglected. The non-dimensional form of the magnetic induction equation is introduced to highlight the relative weight of the terms composing it. The following non-dimensional variables are first defined: 𝒙∗ =
𝒙 ∗ 𝑩 ∗ 𝒕𝑽𝟎 ∗ 𝑽 ,𝑩 = ,𝒕 = ,𝑽 = 𝑳 𝑩𝟎 𝑳 𝑽𝟎
(3)
where 𝐿, 𝑉0 and 𝐵0 are, respectively, a characteristic length, a characteristic velocity and the imposed magnetic field intensity. The non-dimensional form of the magnetic induction equation is thus:
𝝏𝑩∗ 𝟏 − 𝜵∗ × (𝑽∗ × 𝑩∗ ) + 𝜵∗ × (𝜵∗ × 𝑩∗ ) = 𝟎 𝝏𝒕∗ 𝑹𝒆𝒎
(4)
The ratio of the convection of the magnetic field to its diffusion is expressed by the non-dimensional magnetic Reynolds number 𝑅𝑒𝑚 , defined as: 𝑹𝒆𝒎 =
𝒖𝟎 𝑳 𝜼
(5)
Within the boundaries of the low magnetic Reynolds number approximation, the convective effects of the velocity on the magnetic field are negligible compared to its diffusion. Thus, when 𝑅𝑒𝑚 ≪ 1, equation (4) reduces to a Laplacianlike one, stating that the magnetic field is diffused in the domain and vanishes as it approaches the external boundary. This means that the induced magnetic field is negligible and the total magnetic field can be approximated by the imposed one [28-29]. The electromagnetic body force 𝒇𝑳 is obtained by the Ohm’s law: 𝒇𝑳 = 𝒋 × 𝑩 = 𝝈 · (𝑬 + 𝑽 × 𝑩𝟎 ) × 𝑩𝟎
(6)
In this case, system (1) reduces to the compressible Navier-Stokes equations with additional MGD source terms for the electromagnetic body force and Joule heating:
5
𝝏𝝆 + 𝜵 ⋅ (𝝆𝑽) = 𝟎 𝝏𝒕 𝝏𝝆𝑽 + 𝜵 ⋅ (𝝆𝑽𝑽 + 𝒑) − 𝜵 ⋅ 𝝉 = 𝝈 · (𝑬 + 𝑽 × 𝑩𝟎 ) × 𝑩𝟎 𝝏𝒕
(7)
𝝏𝝆𝒆𝒕 + 𝜵 ⋅ ((𝝆𝒆𝒕 + 𝒑)𝑽) − 𝜵 ⋅ (𝑽 ⋅ 𝝉) = 𝝈 · (𝑬 + 𝑽 × 𝑩𝟎 ) · 𝑬 𝝏𝒕
For steady-state flows, system (7) can be written in compact form as: 𝜵 ⋅ (𝑭𝑨 (𝑸) − 𝑭𝑽 (𝑸, 𝜵𝑸)) = 𝑺𝑳
(8)
where 𝑄 the vector of conservative variables is: 𝑸 = [𝝆, 𝝆𝑽, 𝝆𝒆𝒕 ]𝑻
(9)
𝑆 𝐿 = [0, 𝝈 · (𝑬 + 𝑽 × 𝑩0 ) × 𝑩0 , 𝑬 · 𝒋]𝑇
(10)
and 𝑆 𝐿 the source term is:
The inviscid flux vector, the viscous flux vector and the electromagnetic source terms are expressed in Cartesian coordinates as: 𝜌𝑣 𝜌𝑢 𝜌𝑤 𝜌𝑢𝑣 𝜌𝑢𝑤 𝜌𝑢2 + 𝑝 𝜌𝑣𝑤 𝜌𝑢𝑣 𝜌𝑣 2 + 𝑝 𝐅𝑥𝐴 = , 𝐅𝑦𝐴 = , 𝐅𝑧𝐴 = 𝜌𝑢𝑤 𝜌𝑤 2 + 𝑝 𝜌𝑣𝑤 {𝑢(𝜌𝑒 𝑡 + 𝑝)} {𝑤(𝜌𝑒 𝑡 + 𝑝)} {𝑣(𝜌𝑒 𝑡 + 𝑝)} 0 0 0 𝜏𝑦𝑥 𝜏𝑥𝑥 𝜏𝑧𝑥 𝜏𝑦𝑦 𝜏𝑥𝑦 𝜏𝑧𝑦 𝐅𝑥𝑉 = , 𝐅𝑦𝑉 = , 𝐅𝑧𝑉 = 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑧𝑧 {𝐕𝜏𝑥 + 𝑞𝑥 } {𝐕𝜏𝑧 + 𝑞𝑧 } {𝐕𝜏𝑦 + 𝑞𝑦 } 0 2 2 𝐸𝑦 𝐵0𝑧 − 𝐸𝑧 𝐵0𝑦 + 𝑤𝐵0𝑥 𝐵0𝑧 + 𝑣𝐵0𝑥 𝐵0𝑦 − 𝑢(𝐵𝑜𝑦 + 𝐵0𝑧 ) 2 2 𝐿 𝐸 𝐵 − 𝐸 𝐵 + 𝑢𝐵 𝐵 + 𝑤𝐵 𝐵 − 𝑣(𝐵 + 𝐵 ) 𝑧 0𝑥 𝑥 0𝑧 0𝑦 0𝑥 0𝑦 0𝑧 𝑜𝑥 𝑆 =𝜎 0𝑧 2 2 𝐸𝑥 𝐵0𝑦 − 𝐸𝑦 𝐵0𝑥 + 𝑢𝐵0𝑧 𝐵0𝑥 + 𝑣𝐵0𝑧 𝐵0𝑦 − 𝑤(𝐵𝑜𝑥 + 𝐵0𝑦 ) {(𝐸𝑥 + 𝑣𝐵0𝑧 − 𝑤𝐵0𝑦 )𝐸𝑥 + (𝐸𝑦 + 𝑤𝐵0𝑥 − 𝑢𝐵0𝑧 )𝐸𝑦 + (𝐸𝑧 + 𝑢𝐵0𝑦 − 𝑣𝐵0𝑥 )𝐸𝑧 }
(11)
(12)
(13)
2.2. Electric field The current continuity equation is obtained by taking the divergence of the Ampere-Maxwell law:
6
(14)
𝜵 ⋅ (𝜵 × 𝑩) = 𝝁𝟎 𝜵 ⋅ 𝒋 = 𝟎 where 𝜇0 is the permeability of free space. The electric potential 𝜙 is introduced through 𝑬 = −𝜵𝜙, yielding
(15)
𝜵 ⋅ [𝛔 · (−𝜵𝜙 + 𝑽 × 𝑩)] = 0
a scalar equation.
3. Numerical Methodology 3.1. Flow field The edge-based Galerkin formulation was first introduced by Selmin [26]. In this formulation, a summation over pairs of nodes is performed during assembly, such that solution-dependent quantities are factored out from quantities dependent on the spatial discretization. Consequently, the edge coefficients associated with the geometrical features of the mesh can be computed in the pre-processing phase of the solution process, rather than at every iteration [30]. The weak-Galerkin formulation of equation (8) is written: ∑ ∫ 𝛁𝑊𝑖 ⋅ (𝐅𝐴 − 𝐅 𝑉 ) 𝑑𝑉𝑒 + ∑ ∫ 𝑊𝑖 𝐧 ⋅ (𝐅𝐴 − 𝐅 𝑉 ) 𝑑𝐴𝑒 = ∑ 𝑊𝑖 S 𝐿 𝑒∈𝐸𝑖 𝑉𝑒
𝑒∈𝐹𝑖 𝐴𝑒
(16)
𝑒∈𝐸𝑖
where 𝑊𝑖 is the i-th linear test function, and where 𝐸𝑖 and 𝐹𝑖 are the set of elements/boundary faces sharing the i-th vertex. For ease of notation, the contribution from the boundary integrals is ignored from this point on. Cast in an edge-based fashion, equation (16) becomes:
∑ 𝜼𝒊𝒋 ⋅ 𝒋∈𝑲𝒊
𝑭𝑨𝒋 + 𝑭𝑨𝒊 𝟐
− 𝝌𝒊𝒋 ⋅
𝑭𝑨𝒋 − 𝑭𝑨𝒊 𝟐
+ ∑ ∫ 𝜵𝑾𝒊 ⋅ 𝑭𝑽 𝒅𝑽 = 𝑳𝒊 𝑺𝑳𝒊 + 𝑪𝑳𝒊
(17)
𝒆∈𝑬𝒊 𝑽𝒆
where 𝐾𝑖 is the set of nodes connected to the i-th node by an element. The edge coefficients on the left-hand side are 𝜼𝒊𝒋 = ∑ ∫ 𝑾𝒊 𝜵𝑾𝒋 − 𝑾𝒋 𝜵𝑾𝒊 𝒅𝑽
𝝌𝒊𝒋 = ∑ ∫ 𝑾𝒊 𝑾𝒋 𝒏𝒅𝑨
𝒆∈𝑬𝒊 𝑽𝒆
(18)
𝒆∈𝑬𝒊 𝑨𝒆
and 𝐶𝑖𝐿 is the correction term introduced by the edge-based discretization, defined as: 𝑪𝑳𝒊 = ∑ 𝑯𝒊𝒋 𝒋∈𝑲𝒊
𝑺𝑳𝒋 − 𝑺𝑳𝒊 𝟐
(19)
7
and 𝐻𝑖𝑗 and 𝐿𝑖 are the consistent matrix correction term and the lumped mass matrix defined as:
𝑯𝒊𝒋 = ∑ ∫ 𝑾𝒊 𝑾𝒋 𝒅𝑽
𝑳𝒊 = ∑ ∫ 𝑾𝒊 𝒅𝑽
𝒆∈𝑬𝒊 𝑽𝒆
(20)
𝒆∈𝑬𝒊 𝑽𝒆
Note that the application of both 𝐿𝑖 and 𝐻𝑖𝑗 , as shown in the right-hand side of equation (17) is equal to the multiplication of the consistent mass matrix with the source vector. To stabilize the advective term, the vector of inviscid fluxes 𝑭𝑨 is replaced by Roe numerical fluxes 𝑭𝑅𝑜𝑒 evaluated at edges midpoint [31]:
∑ 𝑭𝑹𝒐𝒆 (𝑸𝒊 , 𝑸𝒋 , 𝜼𝒊𝒋 ) − ∑ 𝝌𝒊𝒋 ⋅ 𝒋∈𝑲𝒊
𝑭𝑨𝒋 − 𝑭𝑨𝒊 𝟐
𝒋∈𝑲𝒊
+ ∑ ∫ 𝜵𝑾𝒊 ⋅ 𝑭𝑽 𝒅𝑽 = 𝑳𝒊 𝑺𝑳𝒊 + 𝑪𝑳𝒊
(21)
𝒆∈𝑬𝒊 𝑽𝒆
Second-order accuracy is obtained by adopting a MUSCL reconstruction of the primitive variables at the edges midpoint and monotony is ensured by a 1D van Albada slope limiter [32]. The viscous fluxes are discretized through the standard continuous Galerkin method and assembled in an edge-based fashion, naturally resulting in a secondorder discretization [33]. 3.2. Electric field The weak-Galerkin formulation of the current continuity equation is written: ∑ ∫ 𝛁𝑊𝑖 ⋅ (𝜎 · 𝜵𝜙) 𝑑𝑉𝑒 − ∑ ∫ 𝑊𝑖 𝐧 ⋅ (𝜎 · 𝜵𝜙) 𝑑𝐴𝑒 + ∑ ∫ 𝑊𝑖 𝛁 ⋅ (𝜎 · 𝑭𝐵 )𝑑𝑉𝑒 = 0 𝑒∈𝐸𝑖 𝑉𝑒
𝑒∈𝐹𝑖 𝐴𝑒
(22)
𝑒∈𝐸𝑖 𝑉𝑒
where 𝑭𝐵 is defined as VⅹB. Cast in an edge-based fashion, equation (16) becomes: ∑ 𝝈𝒊𝒋 ∶ 𝜼𝒊𝒋
𝑩 𝑭𝑩 𝒋 + 𝑭𝒊
𝟐
𝒋∈𝑲𝒊
− 𝝈𝒊𝒋 ∶ 𝝌𝒊𝒋
𝑩 𝑭𝑩 𝒋 − 𝑭𝒊
𝟐
+ 𝝈𝒊𝒋 ∶ 𝑫𝒊𝒋 (𝝓𝒋 − 𝝓𝒊 ) + 𝝈𝒊 ∶ 𝝃𝒊 𝜵𝝓𝒊 = 𝟎
(23)
where 𝑫𝑖𝑗 is the stiffness matrix associated with the i-th and j-th nodes and 𝝃𝑖 is defined only on the boundary:
𝑫𝒊𝒋 = ∑ ∫ 𝜵𝑾𝒊 𝜵𝑾𝒋 𝒅𝑽 𝒆∈𝑬𝒊 𝑽𝒆
𝝃𝒊 = ∑ ∫ 𝑾𝒊 𝒏𝒅𝑨.
(24)
𝒆∈𝑬𝒊 𝑨𝒆
3.3. Solution strategy The solution strategy makes use of a pseudo-transient continuation method where the original steady state problem is cast as a pseudo-unsteady one: 𝐴 ∆𝑄𝑛+1 = −𝑅(𝑄𝑛 ) 𝑛 = 0,1,2, …
(25)
8
𝑅 is the residual of equation (21) and matrix 𝐴 is: 𝑛 1 𝜕𝑅̃ 𝐴=[ 𝐼+ ] ∆𝜏 𝜕𝑄
(26)
where 𝑅̃ is an approximation of 𝑅 that does not include the MUSCL reconstruction. ∆𝜏 𝑛 is a pseudo-time step chosen as to locally satisfy the Courant-Friedrichs-Lewy (CFL) stability condition [34]. Note that small values of ∆𝜏 𝑛 increase the diagonal dominance of the system matrix, making it easier to invert. As ∆𝜏 𝑛 increases the pseudounsteady problem reverts to the exact Newton method. Equation (25) is solved by applying the Generalized Minimal Residual (GMRES) method with an Incomplete Lower Upper (ILU) factorization of the matrix 𝐴 as a right preconditioner [35]. 4. Numerical Results The proposed numerical methodology has been validated over the following three-dimensional test cases. These three test cases are respectively inviscid at Mach 5, viscous at Mach 4.75 flows over a sphere and an Apollo capsule reentry situation at Mach 13.26 and angle of attack 18.6°. The Hall effect is not considered in this study. 4.1. Mach 5 Inviscid Flow past a Hemisphere In the first test case, a Mach 5 inviscid flow past a hemisphere is simulated under the effect of an imposed magnetic dipole. The hemisphere radius is 𝑟0 = 10 𝑚𝑚. The properties of the free stream are 𝑇∞ = 100 𝐾, 𝑝∞ = 1587 𝑃𝑎, 𝜌∞ = 0.0553 𝑘𝑔 𝑚−3 , 𝜎∞ = 794 Ω−1 𝑚−1 , 𝑉∞ = 1.0022 × 103 𝑚 𝑠 −1 The electric conductivity is assumed to be uniform all over the domain, and equal to its free stream value. The Cartesian components of the imposed magnetic field are expressed as:
𝐵𝑥 =
3𝑀𝑥𝑧 3𝑀𝑦𝑧 𝑀(3𝑧 2 − 𝑟 2 ) , 𝐵 = , 𝐵 = 𝑦 𝑧 𝑟5 𝑟5 𝑟5
where: 𝑀=
𝐵0 𝑟0 3 2
and 𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2
the origin of the Cartesian system of coordinates being at the center of the hemisphere. Numerical results are presented in terms of the non-dimensional magnetic interaction number:
𝑆𝑡 =
𝜎∞ 𝐵0 2 𝑟0 𝜌∞ 𝑉∞
(27)
9
This non-dimensional group, also known as Stuart number, measures the ratio of electromagnetic forces to inertial ones and it is varied here by setting the value of the imposed magnetic field intensity 𝐵0 . The applied magnetic field in the center of the sphere is provided in Table 1. The CFL number starts at 1, and is exponentially increased to 50 within 100 Newton iterations. 𝑆𝑡
0
1
2
3
4
5
6
𝐵0 [𝑇]
0.000
2.642
3.737
4.577
5.285
5.909
6.472
Table 1. Strength of the applied dipole for the inviscid flow over a sphere. Grid independence study is performed using a coarse mesh containing 171,105 nodes and 163,840 hexahedral bilinear elements (G2, Figure 1) and a fine mesh containing 1,339,585 nodes and 1,310,720 elements (G3). In Figure 2, the static temperature profiles along the stagnation line are plotted for 𝑆𝑡 = 0 (no imposed magnetic field) and 𝑆𝑡 = 6. While a finer mesh (G3) is more effective in terms of shock sharpness, the discrepancy between the two meshes is insignificant. We therefore proceed with the coarser mesh (G2) for the sake of computational efficiency. As described in Section 3.1, the edge-based discretization introduces a correction term 𝐶𝑖𝐿 . In Figure 3, the impact of 𝐶𝑖𝐿 is investigated by plotting the static temperature profiles across the shockwave for 𝑆𝑡 = 6 . No observable difference is found when neglecting the correction term. In Figures 4, 5 and 6, the contours of static temperature, pressure and Mach number are shown for 𝑆𝑡 = 0 and 𝑆𝑡 = 6. Under the influence of an increasing electromagnetic body force, the bow shock is significantly pushed away from the hemisphere surface, thus, increasing the shock standoff distance. To study the effects of the induced electric field, the electric potential, as well the three components of the electric field is exhibited in Figure 7.The electric potential is non-uniform only in regions close to the surface of the sphere and the resulting electric field is therefore non-zero only in these regions. The induced electric field generally enhances MHD heat-shield. Figure 8 reports the shock standoff distance, made non-dimensional with r0, as a function of the magnetic interaction number. The results obtained with the standard low-magnetic Reynolds number approximation and the simplified version (neglecting the induced electric field) are compared with the ones available from the literature [36-38]. Results of the simplified low-magnetic Reynolds number approximation are in reasonable agreement with Poggie and Gaitonde [38]. When 𝑆𝑡 is increased from 0 to 6, the shock standoff distance witnesses an increase of 70%. This is as expected since both are performed without considering the induced electric field. When the induced electric field is accounted for, the shock is further pushed away from the sphere. An 87% increase in shock
10
standoff distance is observed for 𝑆𝑡 = 6. As the imposed magnetic field is strengthened, this enhancement is more significant.
Figure 1. Computational mesh for inviscid flow past a hemisphere (G2).
Figure 2. Comparison of static temperature across the shock wave with different meshes, expressed in Kelvins
11
Figure 3. Impact of correction term 𝐶𝑖𝐿 ; static temperature profiles across the shock wave with different meshes, expressed in Kelvins
\
Figure 4. Contour plots of static temperature (K) for 𝑆𝑡 = 0 and 𝑆𝑡 = 6.
12
Figure 5. Contour plots of pressure (Pa) for 𝑆𝑡 = 0 and 𝑆𝑡 = 6.
Figure 6. Contour plots of Mach number for 𝑆𝑡 = 0 and 𝑆𝑡 = 6.
Figure 7. Contour plots of electric potential (V) and the induced electric field (V/m) for 𝑆𝑡 = 6.
13
Figure 8. Non-dimensional shock standoff distance in terms of the magnetic interaction number, St. The convergence of the obtained numerical results has been assessed. As an example, Figure 9 shows the 𝐿2 norm of the residual 𝑅, in terms of the Newton iterations for 𝑆𝑡 = 6. After 78 iterations the residual drops approximately 4 orders of magnitude, which is more than sufficient for inviscid flows. This simulation used 4 processors and the CPU hours is 0.64.
Figure 9. 𝐿2 norm of residual in terms of Newton iterations 𝑆𝑡 = 6. 4.2. Mach 4.75 Viscous Flow past a Sphere The second test case is a Mach 4.75 laminar flow of Argon past a sphere. The radius of the sphere is 𝑟0 = 38.1 𝑚𝑚 and the hybrid mesh consisting of 196,686 tetrahedral elements and 224,074 nodes is shown in Figure 10. The inset in Figure 10 exhibits the prism layer of 365,760 elements used to resolve the boundary layer. It was judged that such a mesh was fine enough to demonstrate the effect of magnetic number on standoff distance. For more quantitative
14
answers, mesh optimization such as in [41], demonstrated in the following section, could be used. The free stream conditions are 𝑇∞ = 1100 𝐾, 𝑝∞ = 27.8 𝑃𝑎, 𝜌∞ = 1.214 × 10−4 𝑘𝑔 𝑚−3 , 𝑉∞ = 3 × 103 𝑚 𝑠 −1 The electric conductivity of Argon is set to [39]: 𝜎(𝑇) = 3.34 × 10−10
𝛼
(28)
𝛽√𝑇
where: 𝛼 = 0.00623 and 𝛽 = 5 × 10−17 𝑐𝑚2 The free stream electric conductivity is thus 𝜎∞ = 1254.78 Ω−1 𝑚−1 The imposed magnetic field is the same as in the previous test case. The Reynolds number based on the sphere radius is 𝑅𝑒 = 148 and the laminar flow assumption is well justified. The CFL number starts at 0.01 and is exponentially increased to 20 in 50 Newton iterations. An isothermal boundary condition is imposed at the wall by setting 𝑇𝑤 = 300 𝐾. As in the previous test case, numerical results are presented in terms of the non-dimensional magnetic interaction number. Figures 11, 12 and 13 show the contours of static temperature, pressure and Mach number for 𝑆𝑡 = 0 and 𝑆𝑡 = 6. The applied magnetic field in the center of the sphere is provided in Table 2. As for the inviscid case, the bow shock is significantly pushed away from the hemisphere surface under the influence of an increasing electromagnetic body force. 𝑆𝑡
0
1
2
3
4
5
6
𝐵0 [𝑇]
0.000
0.0873
0.1234
0.1512
0.1746
0.1952
0.2138
Table 2. Strength of the applied dipole for the viscous flow over a sphere.
Figure 14 exhibits the contours of the electric potential and the induced electric field. Similar to the test case of the preceding section, the non-uniform electric potential is observed only in regions adjacent to the sphere. The resulting electric field is therefore nontrivial only in these regions.
15
Figure 15 reports non-dimensional shock standoff distance as a function of 𝑆𝑡. When neglecting the induced electric field, the shock standoff distance varies linearly from a value of 0.24, for 𝑆𝑡 = 0, to a value of 0.48, for 𝑆𝑡 = 6, which means a 75% increase. If the induced electric field is taken into account, the increase in shock standoff is more salient and a 100% increase is observed for 𝑆𝑡 = 6. To the best of the authors’ knowledge, no results are available in the literature to compare with for this case. In Figure 16 the non-dimensional peak heat flux is plotted against 𝑆𝑡. The peak heat occurs at the stagnation point and is made non dimensional with the peak heat flux obtained at 𝑆𝑡 = 0. A quasi linear decrease rate is observed up to 𝑆𝑡 = 6, where a value of 0.87 is obtained for a 13% decrease in the peak heat flux.
Figure 10. Computational mesh used for the simulation of a viscous flow past a sphere.
Figure 11. Contour plots of temperature (K) at St = 0 and St = 6.
16
Figure 12. Contour plots of pressure (Pa) at St = 0 and St = 6.
Figure 13. Contour plots of Mach number at St = 0 and St = 6.
17
Figure 14. Contour plots of electric potential (V) and the induced electric field (V/m) for 𝑆𝑡 = 6.
Figure 15. Non-dimensional shock standoff distance in terms of the magnetic interaction number, St.
18
Figure 16. Non-dimensional peak heat flux in terms of the magnetic interaction number, St. Convergence has also been assessed for this viscous test case. As an example, Figure 17 shows the 𝐿2 norm of the residual 𝑅, in terms of Newton iterations for 𝑆𝑡 = 6. A reduction of 5 orders of magnitude in the residual is achieved after 130 iterations, which is satisfactory for a low Reynolds number flow. 8 processors were employed by this simulation and the consumption of CPU hours is 6.2.
Figure 17. 𝐿2 norm of residual in terms of Newton iterations. 4.3. Apollo-like Capsule Re-entry The third test case is the viscous flow over an Apollo command module-like (AS-202) re-entry situation. In this capsule and as shown in Figure 18, the outer mold line (OML) consists of a spherical section (forebody in the flow-
19
oriented nomenclature) of radius of curvature of 4.694 𝑚 with a shoulder radius of 0.196 𝑚, and an after body consisting of a 33 degrees conical section blunted to a 0.231 𝑚 radius at the aft end. The maximal diameter of the capsule is 3.912 𝑚 and the axial length including the TPS material is 3.431 𝑚. The origin of the Cartesian coordinates system is at the intersection between the spherical section and the axial length axis x.
x dipole
Figure 18. Schematic drawing of the outer mold line of AS-202 taken from [40]. The chosen re-entry condition has an angle of attack of 18.6° and a Mach number of 13.26. The capsule temperature is set to 𝑇𝑤 = 8500 𝐾. The air being considered a thermally perfect gas, the free-stream conditions are as follows: 𝑇∞ = 268 𝐾, 𝑝∞ = 61.53 𝑃𝑎, 𝜌∞ = 8 × 10−4 𝑘𝑔 𝑚−3 , 𝑉∞ = 4350 𝑚 𝑠 −1 The electric conductivity is uniform and set to: 𝜎 = 794.00 Ω−1 𝑚−1 [38] The imposed magnetic field is the same as in the two first test cases. The origin of the Cartesian system of coordinates (𝑥, 𝑦, 𝑧) in which the magnetic field components are expressed is placed at (𝑥 = 1.0, 𝑦 = 𝑧 = 0.0) such that 𝑟 = 0, 𝐵𝑥 = 𝐵𝑦 = 𝐵𝑧 = 0 at that point. The Reynolds number based on the capsule length axis is 𝑅𝑒 = 800,000 .
20
Turbulence is modeled through Spalart-Allmaras turbulence model with a free-stream ratio of Eddy turbulent viscosity to laminar one equal to 10−5 . The geometry of the test case being complex with expected more challenging flow characteristics, a mesh adaptation methodology was used and gave very satisfactory results. This methodology seeks numerical solutions on anisotropic adapted unstructured grids. Based on a posteriori estimates, the approach successively modifies the grid through mesh movement, edge swapping, coarsening and refinement, according to eigenvalues and eigenvectors extracted from the Hessian (truncation error) of the evolving solution. For details of the methodology on structured and unstructured grids, for inviscid and viscous flows, one can consult Habashi et al. [41]. The paper convincingly demonstrates that one can iteratively reach a uniquely defined optimal grid starting from any initial mesh. Herein, the solution adaption is carried out for only 4 cycles, with the adaptation criteria being density 𝜌, pressure 𝑝, temperature 𝑇, and turbulent viscosity 𝜇𝑡 . As an example of the mesh optimization, Figure 19 shows the initial mesh and the mesh after four adaptation cycles, for a magnetic interaction number 𝑆𝑡 = 0. The initial original mesh is composed of 909,713 nodes, with 526,273 tetrahedral elements and 1,626,088 prisms, whereas the adapted one is composed of 1,145,647 nodes with 1,527,658 tetrahedral elements and 1,732,112 prisms. One can notice the vast improvement in the solution, with only a 10% nodes increase but with “directionality” taken into consideration, as opposed to an unguided mesh refinement scheme in which points are increased in all directions.
21
Figure 19. Initial mesh (left) and optimal one (right) after 4 adaptation cycles, 𝑆𝑡 = 0. In this case (𝑆𝑡 = 0), CFL number starts at 0.01 and is progressively increased to 0.5 in 1,500 iterations. Figure 20 shows the static temperature contours for 𝑆𝑡 = 0 obtained from the initial mesh (left) and the adapted one (right). It is clear that after mesh adaptation the shock-capturing is considerably improved since the mesh is realigned with the shock adding grid points in regions of higher truncation error.
22
Figure 20. Contour plots of temperature for St = 0, with the initial mesh (left) and with the adapted one (right), expressed in Kelvins. In Figure 21, the adapted meshes after 4 cycles are contrasted for 𝑆𝑡 = 0 (1,145,647 nodes, 1,527,658 tetrahedral elements and 1,732,112 prisms) and 𝑆𝑡 = 6 (𝐵0 = 0.1622 𝑇), demonstrating how the mesh adaptation takes into consideration the local physics. Not only is the shock location captured well, but also the larger wake structure generated by the magnetic dipole.
23
Figure 21. Adapted meshes after 4 adaptation cycles for 𝑆𝑡 = 0 (left) and 𝑆𝑡 = 6 (right). In this case (𝑆𝑡 = 6), CFL number starts at 0.01 and is increased to 0.1 in 1,500 iterations. Figure 22 shows the static temperature contours for the same 𝑆𝑡 = 6 obtained with the initial mesh (left) and with the adapted one (right). Figure 23 exhibits the contours of pressure and Mach number for 𝑆𝑡 = 6 with the adapted mesh.
Figure 22. Contour plots of temperature for St = 6, with the initial mesh (left) and with the adapted one (right), expressed in Kelvin.
24
Figure 23. Contour plots of pressure (Pa, left) and Mach number (right) for St = 6. In Figure 24 the distribution of the static pressure across the shock wave is shown. The usefulness and the efficiency of the mesh adaptation methodology are clearly put into evidence in this figure.
Figure 24. Static temperature across the shock wave, expressed in Kelvins.
25
Figure 25 reports non-dimensional (with respect to the forebody diameter) shock standoff distances as a function of 𝑆𝑡. When neglecting the induced electric field, they vary linearly from 0.1176, for 𝑆𝑡 = 0, to 0.1503, for 𝑆𝑡 = 6, which means an increase of more than 27%. For the standard low-magnetic Reynolds number formulation, an increase of 34% is observed.
Figure 25. Non-dimensional shock standoff distance in terms of the magnetic interaction number St for the Apollo capsule. The maximum heat flux on the Apollo capsule structure external surface in terms of the magnetic interactions number 𝑆𝑡 is reported in Figure 26. Between 𝑆𝑡 = 0 and 𝑆𝑡 = 6, the maximum heat flux decreases by more than 16%.
26
Figure 26. Non-dimensional peak heat flux in terms of the magnetic interaction number St for the Apollo capsule. The convergence of the obtained numerical results has also been assessed for this Apollo capsule re-entry test case. As an example, Figure 27 shows the 𝐿2 norm of the residual 𝑅, in terms of the Newton iterations for 𝑆𝑡 = 6 when the adaptive methodology is applied. A reduction of almost 5 orders of magnitude in the residual is achieved after 40,000 iterations which is reasonable for such a stiff problem. This simulation used 64 processors and the consumption of CPU hours is around 9200.
27
Figure 27. 𝐿2 norm of residual in terms of Newton iterations.
5. Conclusions The present paper is part of a large and sustained effort to develop an FEM-edge all speeds compressible turbulent flow solver, suitable for hypersonic flows. The present paper addresses the magnetic field part of the code. The modeling of the problem takes advantage of the low-magnetic Reynolds number approximation, to decouple the flow and the electromagnetic problems. The induced magnetic field is thus negligible and the total magnetic field can be reasonably approximated by the imposed one. The current continuity equation, cast in terms of a scalar electric potential, is solved to obtain the induced electric field. Then, the flow can be modeled through the RANS equations with source terms accounting for the electromagnetic body force and Joule heating. These equations written in conservative form are discretized via an edge-based FE formulation in which shape function coefficients associated with the geometrical features of the mesh can be computed once and for all in the preprocessing phase of the solution process. To stabilize convection dominated regimes, inviscid fluxes are replaced by Roe numerical ones, and second order accuracy is obtained through a MUSCL reconstruction of the primitive variables at the edges midpoints and a standard 1D van Albada slope limiter. The solution strategy is based on a pseudo-transient
28
continuation method where the original steady-state problem is cast as a pseudo-unsteady one, and the linear systems of equations are solved by a GMRES-ILU method. Solutions have been obtained for three test cases: inviscid and viscous flows over a sphere, and viscous flow over an Apollo re-entry capsule geometry. Numerical results show that in all cases the flow features are changed by the imposed magnetic field. When the Stuart number 𝑆𝑡 gradually grows from 0 to 6, the inviscid and viscous flows over a sphere witness an increase in shock standoff distance of more than 70%, and the Apollo capsule problem also sees an increase of 27%. Moreover, it is put into evidence that the peak heat flux decreases as well (13% for the viscous flow over a sphere problem and 16% for the Apollo problem). For the Apollo capsule, grid adaptation proved to be a good strategy to obtain accurate results. For a fixed number of gird nodes (909,713 prior to adaptation and 1,145,647 post adaptation), the sharpness of the shock is substantially improved after adaptation. An investigation on the effects of the induced electric field is also performed. We observe that the induced electric field is nontrivial in regions adjacent to the object and in general enhances the MHD heat-shield phenomena. Therefore, we suggest that the induced electric field must be taken into account by solving the current continuity equation and subsequently taking the gradient of the calculated electric potential. Future efforts will concentrate on the full coupling of the electromagnetic and flow problems. This would help determine the threshold in terms of magnetic Reynolds number beyond which the assumption of negligible magnetic field is no longer justified in hypersonic applications.
6. Acknowledgements The authors would like to acknowledge the financial support of the NSERC-Lockheed Martin-Bell Helicopter Industrial Research Chair for Multi-physics Analysis and Design of Aerospace Systems. The generous availability of supercomputing time from Compute Canada and CLUMEQ Supercomputer Center are much appreciated.
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