A sharp interface Cartesian grid method for viscous

International Journal of Computational Fluid Dynamics

ISSN: 1061-8562 (Print) 1029-0257 (Online) Journal homepage: http://www.tandfonline.com/loi/gcfd20

A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows Pratik Das, Oishik Sen, Gustaaf Jacobs & H. S. Udaykumar To cite this article: Pratik Das, Oishik Sen, Gustaaf Jacobs & H. S. Udaykumar (2017): A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows, International Journal of Computational Fluid Dynamics, DOI: 10.1080/10618562.2017.1351610 To link to this article: http://dx.doi.org/10.1080/10618562.2017.1351610

Published online: 13 Jul 2017.

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Date: 25 July 2017, At: 11:36

INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS,  https://doi.org/./..

A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows Pratik Dasa , Oishik Sena , Gustaaf Jacobsb and H. S. Udaykumara a Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA, USA; b Aerospace Engineering, San Diego State University, San Diego, CA, USA

ABSTRACT

ARTICLE HISTORY

A Cartesian grid-based sharp interface method is presented for viscous simulations of shocked particle-laden flows. The moving solid–fluid interfaces are represented using level sets. A moving least-squares reconstruction is developed to apply the no-slip boundary condition at solid–fluid interfaces and to supply viscous stresses to the fluid. The algorithms developed in this paper are benchmarked against similarity solutions for the boundary layer over a fixed flat plate and against numerical solutions for moving interface problems such as shock-induced lift-off of a cylinder in a channel. The framework is extended to 3D and applied to calculate low Reynolds number steady supersonic flow over a sphere. Viscous simulation of the interaction of a particle cloud with an incident planar shock is demonstrated; the average drag on the particles and the vorticity field in the cloud are compared to the inviscid case to elucidate the effects of viscosity on momentum transfer between the particle and fluid phases. The methods developed will be useful for obtaining accurate momentum and heat transfer closure models for macro-scale shocked particulate flow applications such as blast waves and dust explosions.

Received  April  Accepted  July 

1. Introduction The interaction of shocks with particles is important in various practical applications, such as explosive dispersal of particles in blast waves (Zhang et al. 2001), pneumatic conveyance of particles (Boiko et al. 1997; Bolio and Sinclair 1995), reactive particle dynamics in the nozzle of rocket engines fuelled by solid propellants (Carlson and Hoglund 1964) and dispersion of debris in explosive volcanic eruptions (Anilkumar, Sparks, and Sturtevant 1993). System (i.e. macro-)scale dynamics of a particle cloud interacting with shock waves is typically modelled using homogenised (volume- or phase-averaged) systems of equations (Chang and Kailasanath 2003; Jacobs and Don 2009; Jacobs and Hesthaven 2009) where the details of flows around particles are not resolved. Instead, macroscale descriptions rely on closure terms to model momentum and energy transfer between the fluid and particle phases. Typically, physical experiments are employed to construct closure models to encapsulate the unresolved (i.e. meso-scale) physics (Boiko et al. 1997; Bolio and Sinclair 1995; Carlson and Hoglund 1964; Henderson 1976; Wagner et al. 2012); but physical experiments are expensive to instrument and therefore empirical correlations are limited in parameter space. Particle-resolved numerical simulations of shock-particle interaction can CONTACT H. S. Udaykumar

[email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Shock-particle interaction; sharp interface methods; ghost fluid method; compressible flows; viscous effects

provide an alternate approach for building closure models (Sen et al. 2015). Construction of a robust framework for such particle-resolved simulations of shock-particle interaction – with focus on the effect of viscosity – is the subject of the present paper. Shocked particle-laden flows require the calculation of the motion of embedded particles. Most previous studies (Zółtak and Drikakis 1998; Naiman and Knight 2007; Chaudhuri et al. 2013; Sridharan et al. 2015; Regele et al. 2014; Mehta, Neal et al. 2016) on meso-scale simulation of shock-particle interaction assumed the particles to be fixed in the flow domain. This frozen regime of particles (Mehta, Neal et al. 2016; Rudinger 1964; Parmar, Haselbacher, and Balachandar 2009) in shock-particle interaction applies for the duration over which the shock transits around the particle. During this frozen period, the particles remain static because the time scale of the movement of the particles is larger than time scale of the shock interaction with the particles (Mehta, Neal et al. 2016). However, the velocity of the particles increase as the momentum of the particles equilibrates with the momentum of the post-shock fluid, i.e. during the relaxation and equilibrium periods (Rudinger 1964; Parmar, Haselbacher, and Balachandar 2009) that follow shock passage. These distinct regimes in the motion of the particles, namely the initial frozen and later relaxation and

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equilibrium regimes, have been observed in the experiments performed by Wagner et al. (2012). As the particles are set in motion, the Reynolds number of the flow relative to the particles changes, because of the relative motion of the particle with respect to the carrier fluid. As the particle approaches the fluid velocity, the slip flow Reynolds number approaches zero; the momentum exchange between the particle and the fluid becomes dependent on the particle velocity and viscous effects in the relaxation and equilibration phases. In addition, as the particles move within the cloud, further clustering or rarefaction of the particles in the cloud can result (Boiko et al. 1997; Ling et al. 2012). The drag on a particle in the cloud can therefore change over time because of a combination of shock-induced pressure forces, viscous forces and the evolving volume fraction field inside the cloud. Evolution of the particle clustering in a cloud following shock passage has been shown in several experimental studies (Boiko et al. 1997; Wagner et al. 2012). Therefore, viscous moving boundary simulations are required for an accurate representation of the meso-scale physics of shock-particle interaction. Meso-scale (i.e. particle-resolved) simulations of shock-particle interaction have so far been limited to the inviscid flow assumption (Sridharan et al. 2015; Regele et al. 2014; Mehta, Neal et al. 2016; Sambasivan and Udaykumar 2009; Mehta, Jackson et al. 2016). Inviscid calculations of shock-particle interactions assume that the shock passage time scale is short compared to the viscous time scale, an assumption that holds during the frozen regime of shock-particle interaction (Regele et al. 2014; Mehta, Neal et al. 2016). However, during the relaxation and the equilibrium period, after shock has travelled further downstream, viscous and inertial time scales can become comparable (Regele et al. 2014). The typical size of a particle in a particle cloud is usually of order of 10−3 m to 10−6 m. Because of the small size of the particles, the Reynolds number of the flow is small and the skin-friction drag can be significant. For example, the numerical study by Sun et al. (2005) suggests that the skin-friction drag-coefficient becomes significant as particle sizes decrease to the range of interest for dusty gas flows. Experimental studies (Wegener and Ashkenas 1961) also show that the drag coefficient of a sphere immersed in steady supersonic flow increases as the diameter of the sphere is reduced. From these previous numerical and experimental studies, it is reasonable to infer that viscous forces may contribute significantly to the overall drag response of the small particle interacting with shocks in meso-scale simulations. To study the effects of viscosity in clusters of particles, a robust framework to perform particle-resolved meso-scale simulations is necessary. The development

of numerical methods for meso-scale simulations of particle-laden flows has been an active field of research (Mehta, Neal et al. 2016; Sambasivan and Udaykumar 2009; Bagchi and Balachandar 2003; Xu and Subramaniam 2010; Almomani et al. 2008; Marella and Udaykumar 2004; Tenneti, Garg, and Subramaniam 2011). However, past efforts have focused for the most part on viscous incompressible flows (Bagchi and Balachandar 2003; Xu and Subramaniam 2010; Almomani et al. 2008; Marella and Udaykumar 2004; Tenneti, Garg, and Subramaniam 2011; Ye et al. 1999) or inviscid compressible flows (Sridharan et al. 2015; Regele et al. 2014; Mehta, Neal et al. 2016; Sambasivan and Udaykumar 2009; Mehta, Jackson et al. 2016; Sridharan et al. 2016). The development of frameworks for viscous compressible flows remains relatively unexplored (Chaudhuri et al. 2013; Mizuno et al. 2015; Nouragliev et al. 2003), particularly in the context of immersed moving solids embedded in compressible flow. To trace the particles in a particle-resolved simulation of clouds of particles, a numerical framework is required that allows displacement of the solid–fluid interfaces in the flow domain. In this work, a Cartesian grid-based sharp interface method (Sambasivan and Udaykumar 2009; Udaykumar et al. 2001; Kapahi et al. 2013) is used to compute the motion of particle clusters in the flow field without having to deal with issues associated with grid motion and management. Several researchers (Sambasivan and Udaykumar 2009; Puscas et al. 2015; Forrer and Berger 1999; Schneiders et al. 2016) have made progress towards the development of Cartesian grid-based sharp interface methods for moving boundary problems in compressible flows. Robust methods for viscous compressible flow simulation around complex stationary objects have also been developed (Ghias, Mittal, and Dong 2007; Greene et al. 2016). However, less attention (Mizuno et al. 2015; Nouragliev et al. 2003; Majidi and Afshari 2016) has been paid to the development of numerical frameworks for viscous calculations of moving boundary problems in supersonic flows, particularly for particles set in motion by shock waves. The present work is directed towards the development of a numerical framework for three-dimensional, viscous supersonic flow calculation, where sharp immersed solid interfaces can move freely through a simple fixed Cartesian mesh. The current framework uses a level set-based representation of the immersed boundaries in a background Cartesian grid (Osher and Sethian 1988) allowing for sharp description of the interfaces. However, implementation of boundary conditions at the immersed solid– fluid interfaces can be challenging in this setting because the interfaces do not align with the grid. This aspect demands particular attention for viscous calculations

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where the boundary layer on the solid surface needs to be resolved. A variant of the ghost fluid method (GFM) (Fedkiw et al. 1999) is used in this work to treat the immersed sharp interfaces. Several algorithms for the GFM were proposed in the past (Sambasivan and Udaykumar 2009; Nouragliev et al. 2003; Liu, Khoo, and Yeo 2003) in the context of problems involving shock interaction with solid–fluid interfaces. In previous work, populating the ghost fluid using reflected boundary conditions (RBCs) in conjunction with a Riemann solver was found to yield accurate results for a range of solid–fluid problems, even in the presence of strong shocks (Sambasivan and Udaykumar 2009). However, all of these methods were developed for inviscid flow simulations, which do not present the challenge of capturing a developing boundary layer on the solid surface. Viscous supersonic flow calculations in the sharp interface Cartesian grid framework for moving solid–fluid interfaces have been relatively uncommon (Mizuno et al. 2015; Nourgaliev, Dinh, and Theofanous 2006). Notable early work (Nouragliev et al. 2003; Nourgaliev, Dinh, and Theofanous 2006) used a characteristics-based matching (CBM) method for imposing the no-slip boundary condition at moving solid–fluid interfaces. However, the CBM method requires characteristic decomposition of the flow-variables and discretisation in characteristics space at the interfaces. A comparison of CBM method and the Riemann solver-based RBC method for inviscid flows showed that these methods are capable of generating comparable results (Sambasivan and Udaykumar 2009). In a later work (Kapahi et al. 2013), the RBC method was advanced by using a least-squares approach to reconstruct the fluid velocity field in the vicinity of the solid–fluid interface and to apply the no-penetration condition on the solid–fluid interface. In the current study, this modified least-squares-based RBC method is extended to account for the no-slip boundary condition at the embedded solid–fluid interfaces. Another challenge in the GFM-based sharp-interface framework is to maintain the order of accuracy of the numerical calculation near embedded interfaces. Previous studies (Nouragliev et al. 2003; Majidi and Afshari 2016; Nourgaliev, Dinh, and Theofanous 2006; Houim and Kuo 2013) used ghost layers that were multiple grid cells thick; this ‘thick’ ghost layer was used in constructing stencils for higher-order numerical schemes. In these previous studies, the extension of the flow variables into the ghost region was done using a partial differential equation (PDE) approach for multidimensional extrapolation (Majidi and Afshari 2016). However, PDE-based extrapolation (Aslam 2004) does not preserve the gradients of the original field in the extrapolated region, particularly when shocks impinge on the interface. This problem is exacerbated when thin viscous boundary

3

layers are present on solid surfaces, as expected in the present viscous calculations. In addition, extrapolation of flow variables into multiple layers of ghost points imposes additional computational burden. It also makes interprocessor communication in parallel computing expensive since all the data in a thick ghost layer needs to be communicated to neighbouring processors. The numerical treatment presented in the current study for discretisation at grid points near the solid–fluid interface does not involve extension into multiple layers of ghost cells in the solid particle; the discretisation also explicitly accounts for the no-slip boundary condition at the location of the interface. The current method preserves the order of accuracy of the numerical scheme at the interface-adjacent grid points while using only one layer of ghost cells (restricted only to cells through which the embedded interfaces pass). This is achieved using onesided high-order calculations of gradients at computational nodes in the vicinity of the solid surface. Extensive validation of the computed interfacial flow quantities (e.g. velocity profile in boundary layer, wall normal pressure, wall shear stresses) shows the effectiveness of the current method. In the following sections, first the governing equations and the numerical framework are presented. The framework is validated against a similarity solution for a compressible boundary layer on a flat plate (Section 3.1). The method is also tested for stationary embedded boundaries by calculating steady supersonic flow over a cylinder and a sphere and comparing with available experimental and numerical results (Sections 3.2 and 3.4). The method is then used to validate the calculations for moving boundary problems by comparing the solutions for the case of lift-off of a cylinder placed in a channel with a numerical benchmark (Sambasivan and Udaykumar 2009; Forrer and Berger 1999) (Section 3.3). Following these validation exercises, the framework is used to compute flows through particle clusters subject to an incident planar shock (Section 3.5). The calculations allow for identification of the differences between viscous and inviscid simulations of particle clouds. Conclusions and future works to extend the techniques are presented in the final section (Section 4).

2. Methods 2.1. Governing equations The governing equations cast in Cartesian co-ordinate system are hyperbolic conservation laws, − → − → − → − → ∂F ∂G ∂H ∂U − → + + + = S. ∂t ∂x ∂y ∂z

(1)

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− → The vector of independent variables U and flux vec→ − − → − → tors F , G , H are ⎛

⎞ ⎞ ⎛ ρ ρu ⎜ ρu ⎟ ⎜ ρu2 + p ⎟ ⎟ ⎟ − − → ⎜ → ⎜ ⎜ ⎟ ⎟ U = ⎜ ρv ⎟, F = ⎜ ⎜ ρuv ⎟, ⎝ρw⎠ ⎝ ρuw ⎠  u ρE + p ρE ⎞ ⎞ ⎛ ⎛ ρv ρw ⎜ ρvu ⎟ ⎜ ρwu ⎟ ⎟ ⎟ − ⎜ − → ⎜ → 2 ⎟ ⎟ ⎜ G =⎜ ⎜ ρv + p ⎟, H = ⎜ ρwv ⎟, ⎝ ρvw ⎠ ⎝ ρw2 + p ⎠   w ρE + p v ρE + p

(7)

(8) (9) (10)

(11)

A third-order ENO-LLF scheme (Shu and Osher 1989) is used for spatial discretisation of the conserved variables. A third-order accurate TVD Runge–Kutta (Gottlieb and Shu 1998) explicit scheme is used for the time integration of the governing equations. A suitable time-step is selected to maintain stability of the solution and is given by

In the above equations, u, v and w are the fluid velocities in the x, y and z direction. ρ is the density of fluid − → and p is the pressure. The source term S accounts for the momentum diffusion, energy dissipation and heat diffu sion effects in governing equations.E =e+ (u2 + v22 +w2 ) is the total specific energy of the fluid and e is the specific internal energy. The components of the viscous stress tensor are calculated as follows:

(2) (3)

 ∂w , ∂z

∂T , ∂x ∂T , ∂y ∂T , ∂z

p = ρ (γ − 1) e.

∂  ∂  uτxx + vτxy +wτxz + uτxy + vτyy +wτyz SE = ∂x ∂y 

∂qx ∂qy ∂qz ∂  + + . + uτxz + vτyz +wτzz − ∂z ∂x ∂y ∂z

 ∂w , ∂z

 2 ∂u ∂v ∂w μ + + . 3 ∂x ∂y ∂z

where, μ is the viscosity of the fluid and Cp is the specific heat of the fluid at constant pressure. The equation of state for an ideal gas is used,

where

2 ∂u ∂v − μ + + 3 ∂x ∂y 

∂v ∂u τxy = μ + , ∂y ∂x 



2 ∂u ∂v ∂v − μ + + τyy = 2 μ ∂y 3 ∂x ∂y 

∂w ∂u τxz = μ + , ∂z ∂x 

∂w ∂v + , τyz = μ ∂z ∂y

−

μCp Pr μCp qy = − Pr μCp qz = − Pr

⎞ 0 ⎜ ∂τxx ∂τxy ∂τxz ⎟ ⎟ ⎜ + + ⎜ ∂x ∂y ∂z ⎟ ⎟ ⎜ ∂τyy ∂τyz ⎟ ∂τxy − → ⎜ ⎟, ⎜ + + S =⎜ ∂y ∂z ⎟ ⎟ ⎜ ∂x ⎜ ∂τxz ∂τyz ∂τzz ⎟ ⎟ ⎜ ⎝ ∂x + ∂y + ∂z ⎠ SE





qx = −

⎛

∂u τxx = 2 μ ∂x

∂w ∂z

The x, y and z components of heat fluxes, i.e. qx , qy and qz are calculated from

and the source vector is



τzz = 2 μ

(4) (5) (6)



 x x2 x2 t = C , , where, u + a 2μ 2k min C ≤ 1 and a is the wave speed.

(12)

Further details of the numerical schemes can be found in previous work (Sambasivan and Udaykumar 2009; Kapahi et al. 2013; Sambasivan and Udaykumar 2010a). 2.2. Sharp interface method Embedded boundaries are represented implicitly using level sets. The value of the level set field φ at any point in space represents the signed normal distance of that point to the corresponding solid–fluid interface. The zero-level contour of the level set field φ provides the exact location of the interface. The level set field is evolved using an advection equation in standard Hamilton–Jacobi form (Sethian and Smereka 2003), ∂φ ∂φ + vi = 0. ∂t ∂xi

(13)

In Equation (13), φ is the level set field corresponding to one or multiple solid objects in the computation domain and vi is the ith velocity component for the motion of the level set field. The level set field

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information is stored only within a narrow band of computational cells around the corresponding zero-level contour to reduce computational cost. The level set advection velocity is obtained by extending the inter→ face velocity (− u  ) into the narrow-band using a PDEbased multidimensional extrapolation method (Aslam 2004). A fifth-order WENO scheme is used for the spatial discretisation of Equation (13), while a fourth-order TVD Runge–Kutta method is employed for the time integration of Equation (13). Details of implementation of the level set method can be found in previous works (Sambasivan and Udaykumar 2009; Kapahi et al. 2013; Sambasivan and Udaykumar 2010a). The level set field is reinitialised after every five time-steps to enforce the  = 1. Further details of the reinitialisacondition |∇φ| tion algorithm can be found in Sambasivan (2010) and Sussman et al. (1998).

2.3. Particle tracking − → → The acceleration (− a CM ), velocity ( U CM ) and location → (− x ) of the centre of mass of the particles are obtained

5

using the following equation:  ∇φ nˆ =   .   ∇φ 

(19)

An explicit Euler scheme is used to update the velocity and position in Equations (14) and (15). The velocity of the solid–fluid interface ( u ) is supplied as the boundary condition for the fluid phase to satisfy no-slip and no-penetration conditions, − → − → u  = U CM .

(20)

Similarly, acceleration of the solid–fluid interface (a ), which appears in the boundary condition for pressure, is obtained from → − → a CM . a=−

(21)

2.4. Boundary conditions

CM

by solving the following equations of motion of the particles in a Lagrangian frame of reference: − → − → d U CM − F CM , =→ a CM = dt Mp → d− x CM − → = U CM , dt

(14) (15)

where, M p is the mass of a single particle. Particles are assumed to be rigid in this study. The resultant force act− → ing on the centre of mass of a particle ( F CM ) is obtained by performing a surface integral of the pressure and viscous shear stresses acting on the solid–fluid interface representing the particle using the following equations:

FP = Fvis =

 S



nˆ x

nˆ y

∂ρ = 0. ∂n

ˆ PndA, ⎡

 τxx nˆ z · ⎣τyx τzx

FCM = FP + Fvis ,

(16) τxy τyy τzy

⎤

τxz τyz ⎦ dA, τzz (17) (18)

where nˆ = [ nˆ x nˆ y nˆ z ] is the unit vector, locally normal to the interface, and it is obtained from the level set field

(22)

A normal force balance at the interface provides the pressure boundary condition, ρs uf t 2 ∂p = − ρs an , ∂n R

 S

For viscous flow calculations, no-slip and no-penetration boundary conditions are enforced at the fluid–solid interfaces. A Dirichlet boundary condition is applied for the velocity components of the fluid. The velocity of the fluid → at the interface (− u f ) is set to the velocity of the solid–fluid → interface representing the embedded rigid object (− u  ). For pressure and density, Neumann boundary conditions are enforced at the solid–fluid interface. The density boundary condition is as follows:

(23)

where uft is the magnitude of tangential component of velocity of the fluid at the interface and an is the magnitude of the normal component of acceleration of the → solid–fluid interface (− a  ). To evaluate the importance of fluid viscosity in shocked particle-laden flows, inviscid simulations are also performed in the current study. For the inviscid calculations, the free-slip, no-penetration velocity boundary conditions at the interface are applied, ufn = u n ,

(24)

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∂ uf t = 0. ∂n

(25)

ufn and uf t are the magnitudes of normal and tangential components of fluid velocity at the interface. In the next section, the implementation of the no-slip boundary conditions is discussed. 2.5. GFM treatment Kapahi et al. (2013) developed a modified GFM for inviscid compressible flows around particles; the slip conditions took the form of Neumann boundary conditions at the interfaces. In the present work, the technique in Kapahi et al. (2013) is modified to enable viscous computations, so that Dirichlet boundary conditions are also implemented at the solid–fluid interface. The implementation proceeds along the following steps. ... Categorisation of computational points Similar to previous works (Udaykumar et al. 2001; Ghias, Mittal, and Dong 2007; Sambasivan and Udaykumar 2010a) the computational nodes in the domain are categorised into four types: fluid points, solid points, ghost points and interfacial points as shown in Figure 1(a). The fluid points (denoted by f followed by numerical values) and the solid points (denoted by s followed by numerical values) carry only the properties of their corresponding (fluid/solid) phases at that location. The solid objects immersed in fluid are considered to be non-deformable. Therefore, flow variables are not computed at the solid points; ghost field values are populated in a single layer of interfacial points in the solid by extending the field from the fluid phase. Ghost points (denoted by g followed by numerical values) refer to points inside the solid that are immediately adjacent to the interface. These points are populated with values of flow variables (density, velocity, pressure and internal energy) extended from the fluid region so that the required boundary conditions are imposed at the solid–fluid interface. The points in the fluid phase which use ghost points in their stencils are categorised as interfacial points (denoted by i followed by numerical values). The next section describes the technique adopted for extending the flow variables into the ghost points. ... Extending the flow variables into ghost points As mentioned above, ghost points are populated with fields extended from the fluid phase and satisfying the interfacial conditions. Figure 1(b) illustrates a typical scenario that arises during such an extension procedure.

Figure . (a) Schematic diagram of the Cartesian grid with an embedded solid object. (b) A schematic diagram of the Cartesian grid to explain the strategy adopted for forming the convex-hull to obtain least-squares approximations of flow variables at the reflected point.

For example, the point i1 in Figure 1(b) is an interfacial point and requires the value at the ghost point g1 in constructing stencils for calculating fluxes of flow variables in the y-direction. Ghost values of the flow variables at the point g1 are obtained by extending the fields from the fluid points closest to the point g1. To perform the extension, first the reflection of the point g1 across the interface boundary, i.e. the point rg1 is obtained. The co-ordinates of rg1 are calculated from xirg1 = xig1 − 2∗φg1 ∗ni ,

(26)

where, xirg1 (i = 1,2 for 2D) are co-ordinates of the point rg1, xig1 (i = 1,2 for 2D) are the co-ordinates for point g1. φg1 is the value of the level set field at point g1 and ni are the components of the normal to the interface. The point rg1 in general does not coincide with a grid point. To obtain the values of flow variables at rg1, first

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the grid-point closest to rg1 is identified. In this example (Figure 1(b)), the point i1 is the grid point in the fluid domain closest to rg1. A point cloud is formed with the grid points in the immediate neighbourhood of the point i1. This point cloud is used to construct a leastsquares approximation of the values of flow variables at the reflected point rg1. Care must be exercised in choosing points to include in the least-squares representation so that the point cloud represents the convex hull for the approximation at rg1. In Figure 1(b), the point cloud forming the convex hull for the least-squares approximation is represented by the points inside the rectangle (denoted by the dashed line). In general, in such a scenario, some of the points in the point cloud are ghost points themselves. This is seen in Figure 1(b) to be the points g1 and g2. Since values of flow variables are initially unknown at these ghost points, they are excluded from the cloud. Instead, the normal projections from these ghost points onto the interface (pg1 and pg2 respectively) are used, which are calculated as xipg1 = xig1 − φg1 ∗ni .

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approach, ψ = a + bˆx + cˆy + dˆxyˆ + eˆx2 + dˆy2 ,

(28)

where xˆ and yˆ are normalised coordinates defined as follows: x − xrg1 , h y − yrg1 yˆ = , h

xˆ =

  N   2  2 h = 0.7 xn − xrg1 + yn − yrg1 .

(29) (30)

(31)

n=1

The values of the flow variables obtained at rg1 from the least-squares approximation are then extended to the ghost point g1 such that a Dirichlet boundary condition is imposed on the interface. This is obtained from uirg1 = 2∗uiboundary − uig1 .

(32)

(27)

No-slip boundary conditions for the velocity components are imposed at these projected points. However, some members of the point cloud (e.g. i2 and i8) are located very close to the interface. This type of positioning of members of the point cloud can result in an ill-conditioned matrix for the least-squares approximation. Inversion of such a matrix leads to erroneous approximation of the flow variables at the point rg1. To circumvent this issue, all the points that are closer to the interface than the reflected point rg1 are excluded from the point cloud. Instead, their projections on the interface (pi2 and pi8) are included in the least-squares cloud and the no-slip boundary condition is explicitly imposed at those projected points. These procedures for selecting points in the least-squares reconstruction ensure a convex hull for the point rg1 and results in strong imposition of Dirichlet boundary conditions at the interface. At this stage a point cloud, as shown in Figure 1(b), surrounding rg1 has been constructed in the fluid domain and consists of the points: i1, pi2, i3, i4, i5, r7, pi8, pg1 and pg2. This point cloud forms the convex-hull in space for least-squares reconstruction of the flow variables. There are a total of N points in the cloud with (xn , yn ) denoting the co-ordinates of the nth cloud point. Let ψ represent a flow variable (such as, velocity, density, pressure, internal energy) in the fluid domain. A multidimensional (2- or 3D, with only 2D scenario illustrated here) polynomial is fit through the point cloud using the least-squares

Once the flow variables at the ghost points are obtained using the algorithm described above, they are used in the stencil for flow calculations at the interfacial points. The procedure for the calculation of fluxes of the flow variable at the fluid points and the interfacial points are discussed in the following section. 2.6. Numerical treatments at the interfacial cells in the fluid phase In the current framework, only the ghost points immediately adjacent to the solid–fluid interface are used in the stencil for flow calculation at the interfacial points. Use of the flow variables extended into the interior ghost points (i.e. ghost points further away from the interface) are avoided in the flux calculation. However, the challenge is to preserve the order of accuracy of the discretisation schemes at the interfacial points without including the extended flow variables beyond the single layer of ghost points. In the following two sections (Sections 2.6.1 and 2.6.2), the numerical treatments for the conservative convective flux terms and the diffusion terms in the governing equation are presented. ... Numerical treatment of the convective terms of the governing equation at the interfacial points In the present framework, a third-order ENO-LLF (Shu and Osher 1989) scheme is used to approximate the → − − → − → values of the fluxes ( F , G , H ) at the cell faces of the fluid cells. However, a different treatment of these

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Figure . Comparison between the (a) distribution of f’ (Equation ()) and (b) distribution of g (Equation ()) against η (Equation ()), obtained from the similarity solution and the simulation of compressible boundary layer over a flat plate at Mach number . and ReL = .

fluxes is employed at the interfacial points. At the interfacial points, a second-order ENO-LLF scheme is used to restrict the reach of the stencil to the first layer of ghost points adjacent to the interface. Higher order flux constructions are possible by using ghost layers more than one layer thick; however, this would require extension of fields from the interior of the fluid into the interior of the solid. Such extensions are typically low-order, computationally expensive and result in deterioration of accuracy of the fluxes at the interfacial points. Consider the point (x0 , y0 ) in Figure 1(a); this is an interfacial point in the fluid domain adjacent to the interface. The solid–fluid interface adjacent to the grid point (x0 , y0 ) lies to the east. Let the location of the interface be represented by . A special treatment is adopted to construct the flux on the east face of this computation cell. The east face of this computational cell is moved to coincide with the location of the solid–fluid interface. The flux − → at the east face of (x0 , y0 ), i.e. F  is obtained by explicitly imposing the no-slip boundary condition at the interface Г so that ⎛ ⎞ 0 ⎜p⎟ ⎜ ⎟ − → ⎟ F=⎜ ⎜0⎟ . ⎝0⎠ 0 

(33)

− → However, the flux at the west face ( F 0 − 12 ) of the grid point (x0 , y0 ) is obtained from a second-order ENO-LLF approximation to avoid inclusion of more than one ghost

Figure . A comparison between pressure, skin-friction and total drag responses of the cylinder immersed in a Mach  ReD =  flow, obtained using different grid resolution.

point in the stencil. Therefore, the derivative of the xcomponent of the conservative flux term in the governing equation at the point (x0 , y0 ) is obtained from the following equation:  − → ∂F ∂x

∼ = [X0 ,Y0 ]

− → − → F  − F 0− 21 x 2

+ x

,

(34)

where x is the distance of the interface (in the xdirection) from the grid point (x0 , y0 ).

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Figure . Convergence study of results obtained from the simulation steady Mach  ReD =  flow over cylinder based on four different grid resolutions.

In the preceding example, the solid–fluid interface is located to the east of the interfacial cell. Following a similar procedure, this algorithm is applied in other cardinal directions as well (e.g. for situations where the interface is located to the north, south, top or bottom of the cell), to obtain fluxes at the cell faces of the interfacial cells. This procedure maintains second-order accuracy at the interfacial points using a single layer of ghost points in the solid. ... Calculation of viscous stress terms In this section, the numerical treatment of the diffusion terms in the governing equations (Equation (1)) is presented. At every time-step, first the gradients of the

9

velocity components (u, v and w) are computed from the velocity field. To accurately compute the velocity gradients in the boundary layer adjacent to the solid surface significantly higher mesh resolution is required than for the inviscid case. The high-resolution mesh requirements near the solid–fluid interfaces can be relaxed if a higher order finite-difference scheme is used. With this in mind, a globally fourth-order accurate finitedifference scheme is used to obtain the gradients of the velocity components. However, implementation of a high-order finite-difference scheme would require a multiple-cell thick ghost fluid layer. Use of high-order finite-difference schemes also increase the burden of communication between processors while performing a parallel computation. To balance these constraints, Greene et al. (2016) used a fourth-order finite-difference scheme to compute the viscous terms and obtained satisfactorily results for the high-gradient velocity field within supersonic boundary-layers. Therefore, as a compromise between accuracy and computational efficiency a fourthorder central difference scheme is used at to compute velocity gradients to obtain diffusive fluxes. One-sided fourth-order finite-difference schemes are used at interfacial points to limit the reach of the stencil to a single layer of ghost points in the solid. Calculation of the velocity gradient at the computational nodes shown in Figure 1(b) to fourth-order accuracy proceeds as follows. Derivatives of the velocity components in the xdirection at point (x-2 , y0 ), i.e. two points away from the interface, are computed using a centred scheme, 

∂ui ∂x

 −2,0

=

−ui0,0 + 8ui−1,0 − 8ui−3,0 + ui−4,0 . (35) 12x

Figure . Temperature contour obtained from simulation of a Mach , ReD =  flow over a circular cylinder on a Cartesian grid with a grid resolution of  points across the cylinder.

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P. DAS ET AL.

Figure . A comparison between (a) distribution of local Mach no. along the centreline behind the cylinder, (b) distribution of local pressure coefficient (Cp (θ )) on the surface of the cylinder and (c) distribution of non-dimensionalised local wall shear-stress (Co (θ )) on the cylinder surface.

At the interfacial points, suitable finite-difference schemes are selected to ensure that only the first ghost point is used in the stencil. For example, for the point (x-1 , y0 ), the following fourth-order central difference scheme is used: 

∂ui ∂x

 −1,0

=

 − ui1,0 ghost value + 8ui0,0 − 8ui−2,0 + ui−3,0 12x

is used for (x0 , y0 ) as given below, 

 ∂ui = ∂x 0,0  3 ui1,0 ghost value + 10ui0,0 −18ui−1,0 +6ui−2,0 −ui−3,0 12 x

.

(36) Unlike (x-1 , y0 ), for the point (x0 , y0 ), use of such a central-difference scheme would extend the derivative calculation to multiple layer of ghost points. Therefore, to restrict the reach of the stencil to only a single ghost point, a fourth-order one-sided finite-difference scheme

.

(37) Values of the viscous stress components are also required at the ghost points for diffusive flux calculations at the interfacial points. This presents some difficulty because in extending the stress field across the interfaces into the ghost points an interfacial boundary condition has to be imposed on the stress field. However, the values

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Figure . Numerical Schlieren of inviscid (a, b, c) and viscous (d, e, f) simulation of cylinder lift-off caused by shock impingement computed at three different time instances: t∗ = . (a & d), t∗ = . (b & e) and t∗ = . (c & f).

of the stress components are unknown at the solid–fluid interfaces. To circumvent this problem with extending the stresses to the ghost point, the values of the stress components at the ghost points are directly computed from the extended velocity field. A one-sided fourth-order finitedifference scheme is used to obtain the spatial derivatives of the velocity field at the ghost points. As an example, the x-derivative of the velocity components at the ghost point (x1 , y0 ) is computed using the following equation: 

∂ui ∂x

=

  1,0



diffusion terms, once again using only a single cell thick ghost layer. Results obtained using the current method is compared the benchmark results in the following section. In this work, a uniform Cartesian mesh is used to solve the benchmark problems, to demonstrate that the current method is suitable for viscous simulations of shocked particle-laden flow in 2D and 3D. Local mesh refinement to adapt to boundary layers and other features the flow field have been used in previous work (Sambasivan and Udaykumar 2010b) with the present computer code but are not deployed for the calculations shown in this paper.

ghost value

25 ui1,0



ghost value

−16ui0,0 +12ui−1,0 − 16ui−2,0 +4ui−3,0 12x

.

(38) Other components of the stress tensor are calculated similarly from spatial derivatives of the velocity components using Equations (2)–(7). Once the velocity gradients and corresponding viscous stresses are obtained, the momentum diffusion terms − → ( S ) in the governing equation are calculated from the spatial derivatives of the stresses. The spatial derivatives of the stress components at the fluid points are computed using a fourth-order central-difference method (as in Equation (35)). However, at the interfacial points in the fluid, a fourth-order one-sided finite-difference scheme (as in Equation (37)) is used to calculate the momentum

3. Results and discussions In this section, the sharp interface method, discussed so far, is used to study several problems of viscous supersonic flows. The method is first validated for stationary and moving boundary problems. Interfacial quantities, such as the boundary layer profiles, wall pressure and the wall shear-stress are compared to analytical results to assess the performance of the current method. Following this, the method is extended for three-dimensional problems: a case of a sphere in a steady supersonic flow is simulated and the steady-state drag coefficient obtained is compared with experimental results. Finally, the interaction of a shock wave with a cloud of particles is studied. The averaged drag-response of the particles in the cloud obtained from the inviscid and viscous simulations are

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compared. These results demonstrate the accuracy of the current approach for viscous compressible flows and evaluate the contributions of viscous effects to the physics of shocked flows in particle clusters.

g + Pr ∗ f g = σ¯ (1 − Pr)( f f ) ,

(44)

where 3.1. Simulation of a laminar compressible boundary layer The flow over a flat plate in a steady laminar compressible flow is simulated for Ma = 0.8 and ReLref (= ρ∞ Uμ∞∞ Lref ) = 1000, where Lref is taken to be the length of the flat plate. ρ∞ and U∞ are the density and velocity of the free-stream flow, respectively. μ∞ is the viscosity of the fluid in the free-stream. The numerically obtained boundary layer profile over the flat plate is compared with similarity solution. The computational domain has a length of x/Lref = 1.25 and height of y/Lref = 0.2. A uniform Cartesian grid of resolution 2000 × 320 points is used, which is equivalent to a grid-resolution of Y = 2.94 along perpendicuy+ lar direction of the flow. A sharp interface represented by the zero level set contour line defines the surface of a flat plate at a height of 0.01 from the bottom of the computational domain. The flat-plate is treated as an adiabatic wall with a no-slip boundary condition. The boundary conditions at the flat-plate if imposed using the current GFM. In the current simulation, compressible boundarylayer profile corresponding to ReL = 1000 is assumed at the inlet of the computation domain. A similarity solution can be developed for this compressible boundary layer profile, as shown in Cohen and Reshotko (1955) and Li and Nagamatsu (1954). The similarity functions fη (η), g(η) and the similarity variable η are defined as  u x, y  , fη (η) = U∞ x, y  H x, y  , g (η) = H∞ x, y U∞ (x) y ∫ ρdy, η = N (x) 0

(39)

(41)

(42)

Application of the preceding transformation of the flow variables on the equations of the compressible boundary layer results in the following coupled ODEs: f + f f = 0,

(γ − 1) Ma2∞ 1+

. γ −1 Ma2∞ 2

(45)

The preceding coupled ODEs are subjected to the following boundary conditions and solved using a fourthorder Runge–Kutta shooting method to obtain the similarity solution for boundary layer,

f (η = 0) = 0,

f (η = ∞) = 1,

(46) (47)

g (η = 0) = 0,

(48)

g (η = ∞) = 1.

(49)

The similarity solution obtained from this preceding calculation is compared with the computed results. Velocity and density profiles along the direction normal to the plate are extracted at the location x/Lref = 1.0 in the flow domain and are scaled in terms of the similarity parameters given by Equations (39) and (40). The boundary layer profile obtained from the simulation is compared with the similarity solution; the computed and analytical solutions are shown superposed in Figure 2. The boundary layer profiles of velocity, fη (η) in Figure 2(a), and enthalpy g(η) in Figure 2(b), are found to be in good agreement with the similarity solution. The momentum and thermal boundary layer thicknesses obtained from the simulation match well with the similarity solution.

(40)

where NNx = 1. Cρ∞ μ∞ U∞

σ¯ =

(43)

3.2. Steady low Reynolds number supersonic flow over a cylinder In this section, steady supersonic flow field around a static cylinder is studied using the current sharp interface method. The Mach number of the freestream flow is 5. The Reynolds number (ReD = ρ∞μU∞∞ D ) of the flow is 250, where D is the non-dimensional diameter of the cylinder and is considered one unit. The computational domain is of (non-dimensional) size 21 × 8 units. Dirichlet and Neumann conditions are applied on the west and the east boundaries while a sponge layer (Bogey, Bailly, and Juvé 2000) is used along with freestream boundary condition on the top and bottom boundaries of the domain to prevent reflection of

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Figure . A comparison of the vorticity contours in the vicinity of the cylinder, obtained from the (a) inviscid and (b) viscous simulations of the cylinder lift-off due to shock-impingement case at time t∗ = ..

Figure . Locus of the centre of the moving cylinder obtained from the simulations of cylinder lift-off due to shock impingement.

Figure . Mesh convergence study for the cylinder lift-off case based on locus of the centre of the cylinder.

pressure waves back into the domain. To study convergence of the current method, four different grid resolutions are used corresponding to 40, 80, 100 and 120 grid points across the cylinder diameter. Grid convergence is shown by comparing the coefficient of drag (CD ) obtained from the different grid resolutions in Figure 3. With grid refinement, the drag forces acting on the cylinder at the steady state converge monotonically. Further analysis is carried out by comparing the

error in the wall-normal pressure (Pw ), wall shear stress (τw ) and drag coefficient (CD ). The L2 errors in these quantities are computed using the following equations:

ErrorPw



  ∫2π Pfine grid − PCoarse grid 2 dθ  0 w w , = 

2  fine grid ∫2π Pw dθ 0

(50)

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P. DAS ET AL.

Figure . The standing bow-shock in the front and the streamlines (coloured with the magnitude of axial velocity component) around a sphere immersed in a steady supersonic (Mach no. = .) low Reynold number ( ReD = ) flow.

Errorτw



  ∫2π τ fine grid − τ Coarse grid 2 dθ  0 w w , = 

2  fine grid ∫2π τ dθ w 0 fine grid

ErrorDrag Coefficient =

CD

(51)

coarse grid

− CD

fine grid

CD

.

(52)

As shown in Figure 4, the wall-normal pressure, wall shear stress on the cylinder surface and the drag coefficient for the cylinder computed using the current method converge with mesh refinement. A second-order rate of convergence of the total drag coefficient of the cylinder is observed in the mesh refinement study. Contours of temperature are shown in Figure 5. The stand-off distance of the bow shock from the front of the cylinder is found to be in agreement with the benchmark simulation (Bashkin et al. 1998). The relatively hightemperature region behind the cylinder in Figure 5 indicates a significant amount of viscous heating on the cylinder surface; the fluid in the boundary layer is advected into the near wake of the cylinder leading to a high temperature region behind the cylinder.

For further validation, the computed results are compared in Figure 6 with a previous numerical result which was obtained using a body-fitted mesh (Bashkin et al. 1998).The local Mach number distribution in the wake of the cylinder obtained using the current framework agrees with the benchmark results (Bashkin et al. 1998) as shown in Figure 6(a). Further analysis of the results obtained from the current numerical study is carried out by computing the local pressure coefficient (Cp (θ )) and the local wall shear stress coefficient (Co (θ )) from, (Pw (θ ) − P∞ ) , 1 ρ U2 2 ∞ ∞ τw (θ ) Re0.5 D Co (θ ) = 1 . 2 ρ U 2 ∞ ∞

Cp (θ ) =

(53) (54)

The prediction of the pressure and wall shear stress distribution on the immersed interface using the current method is in reasonable agreement with the benchmark results (Bashkin et al. 1998), as shown in Figure 6(b,c), respectively. However, the post-processed wall shearstress distribution shown in Figure 6(c) exhibits oscillations. The oscillations seen in the Figure 6(c) are because of the current numerical method used to approximate

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Figure . Velocity contours obtained from (a) inviscid and (b) viscous simulation of low Reynolds number (ReD = ) steady Mach . flow over a sphere at Z/D =  plane.

the wall shear stress at the interface from the computational nodes in the vicinity. In the current numerical method, the solid–fluid interfaces do not align with the computational nodes of the Cartesian mesh. A first-order least-square-based reconstruction of the flow field is used to approximate the wall shear stresses from the computation nodes available in vicinity of the locations at the interface. The number of points in the least-squares field for this approximation at a given location on the interface vary based on the geometry of the interface. Such changes in the geometry of the convex hull of least-square field is the reason for the oscillation seen in the distribution of the wall shear stress. As expected such oscillations are found to decrease with mesh refinement in Figure 6(c). From the results presented in Figures 4–6, it is demonstrated that the current method is capable of predicting the distribution of pressure and wall shear stress on the surface of a cylinder immersed in steady supersonic flow. From Figure 3, it is also evident that in the present case the skin-friction drag contributes about 11% of the total drag on the cylinder.

3.3. 2D simulation of lift-off a cylinder in a shock-tube In this section, the sharp interface method is used to solve a moving boundary problem. The lift-off of a cylinder interacting with a shock is simulated and compared with benchmark results (Sambasivan and Udaykumar 2009; Forrer and Berger 1999). In Sambasivan and Udaykumar (2009) and Forrer and Berger (1999), the computations are limited to inviscid flows only. However, in the current study, the lift-off of a cylinder is simulated for both inviscid and viscous fluid. The trajectory of the centre of the cylinder and the shock dynamics in the flow field obtained from the inviscid and the viscous simulations are compared with each other. The length of the computational domain is selected as the reference length scale and is taken to be 1.0. The height of the domain is 0.2 and the diameter of the cylinder is 0.1 non-dimensional units. The cylinder centre is initially at (0.15, 0.05), i.e. the cylinder is placed close to the bottom wall of the shock tube. The non-dimensional values of the pressure and density of the un-shocked fluid are 1.4 and 1, respectively.

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Figure . Axial velocity distribution in the wake of the sphere immersed in a steady flow of Mach ., ReD =  along the axis of symmetry.

Figure . Axial velocity distribution along radially outward direction from the location X//R = , Y//R = , Z/R =  (considering the centre of the sphere as the origin of the co-ordinate system) on the surface of the sphere.

The Reynolds number of the flow calculated based on the flow conditions behind the travelling shock wave is 240. A shock wave of Ma = 3.0 is located initially at x = 0.08 and is allowed to evolve until t∗ = 0.3. A reflective boundary condition is applied at the top and the bottom edges of the computation domain. Neumann boundary condition is applied at the east and the west edges of the computation domain. In this study, a uniform Cartesian grid is used. Five different grid resolutions are considered, corresponding to 50, 100, 150, 200 and 400 points across

diameter for the grid convergence study. The numerical Schlieren plots presented in Figure 7 for viscous and inviscid simulation are obtained with a grid resolution of 200 points across the diameter of the cylinder. As the flow evolves, the shock is reflected from the bottom wall of the computational domain producing a nonzero lift on the cylinder. The numerical Schlieren fields computed at different time instances (t∗ = 0.0, 0.1, 0.3) are shown in Figure 7 for both the viscous and the inviscid cases. As expected, the shock structures seen in the viscous flow field are diffused compared to those in the inviscid flow field. By comparing the numerical Schlierens in Figure 7(c,f), and the vorticity contours in Figure 8(a,b) at t∗ = 0.3, it is observed that the vortical structures located below the cylinder, near the lower surface of the shock tube, are more diffused in the case of the viscous flow field. In Figure 8(b), it is observed that the flow separates from the cylinder surface in the low Reynolds number viscous simulation; such separation is absent in the inviscid simulation results. In Figure 9, the locus of the centre of the moving cylinder is compared with the benchmark results (Forrer and Berger 1999; Sambasivan and Udaykumar 2010a). The trajectory of the cylinder centre obtained from the current study is in good agreement with the results of previous studies. It is also observed that the lift-off height of the cylinder is somewhat lower in the viscous flow simulation, i.e. viscous effects suppress the lift-off of the cylinder. This effect is modest in the present case, since the length over which the lift-off occurs is small. Thus, although the flow features differ noticeably between the viscous and inviscid cases the differences in the particle motion are not significant for the current cylinder lift-off problem, at least for the duration of the simulation. A convergence study is performed for the above moving boundary problem; the convergence evaluation is based on the errors in tracking the locus of the cylinder centre. The L2 error in the locus of the cylinder is computed from

Errorxci



  ∫T xfine grid − yCoarse grid 2 dt  0 ci ci . = 

2  fine grid T ∫0 xci dt

(55)

The error is seen to monotonically decrease with grid refinement in Figure 10. Results obtained from the simulation of cylinder liftoff caused by shock impingement in a shock tube show that the current GFM is adequate for viscous simulations of moving boundary problems in supersonic flow.

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Figure . Comparison between pressure drag, skin-friction drag and the total drag of a sphere submerged in a Mach no. = ., ReD =  flow.

Figure . Percentage of error (with respect to experimental data) in drag-coefficient of a sphere immersed steady supersonic flow (Mainf. = ., ReD = ) obtained, from three-dimensional viscous simulation.

3.4. 3D simulation of steady Mach 4.1 flow over a sphere (ReD = 612) The sharp interface method is now extended to 3D problems. In this section, viscous and inviscid computations of steady Mach 4.1 flow over a sphere are performed to obtain the drag coefficient for spherical particles and to validate the current framework against experimental results (Wegener and Ashkenas 1961). The Reynolds number for the viscous flow simulation is 612. The diameter of the sphere is taken as the reference length-scale and

is selected to be one non-dimensional unit. The size of the computational domain is 15 × 5 × 5 units. The centre of the sphere is placed at (3, 2.5, 2.5). The west face of the computational domain is designated as the inlet and the east face as the outlet. At the inlet, steady Dirichlet boundary condition is enforced for the flow variables (e.g. density, velocity, pressure and total internal energy), while at the outlet a Neumann boundary condition is applied. A freestream boundary condition corresponding to the Mach 4.1 flow with a sponge layer (Bogey, Bailly, and Juvé 2000) treatment is applied on the north, south, top and bottom faces of the computation domain; the sponge layer prevents pressure waves from reflecting back into the computation domain from those boundaries. The inlet conditions and the freestream conditions applied at the respective edges of the computation domain are as follows: Inlet boundary condition: uinlet =1 Uin f ρinlet ∗ = = 1 ρinlet ρin f Tinlet 1 ∗ Tinlet = 2 = = 1.483 × 10−4 Uin f γ R × Ma2in f Pinlet 1 ∗ Pinlet = = = 0.0426 2 ρin f Uin f γ Ma2in f u∗inlet =

Freestream boundary conditions:

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Figure . Numerical Schlieren computed at t∗ = . from (a) inviscid and (b) viscous simulation of shock-particle-cloud interaction.

ufs =1 Uin f ρfs ρ ∗f s = = 1 ρin f Tf s 1 T f∗s = 2 = = 1.483 × 10−4 Uin f γ R × Ma2in f Pf s 1 P∗f s = = = 0.0426 2 ρin f Uin f γ Ma2in f u∗f s =

Figure . Numerical Schlierens at (a) t∗ = ., (b) t∗ = . and (c) t∗ = . computed from viscous simulation of a moving planar shock-wave (travelling at a velocity of Mach .) interacting with a particle cloud ( particles).

In the above, superscript ‘∗’ denotes nondimensionalised quantities, the subscript ‘inlet’ and ‘fs’ denotes the boundary conditions at the inlet and the freestream, respectively. The subscript ‘inf’ represents the reference flow conditions.

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To study grid dependence, uniform Cartesian grids of three different grid spacing (X = 0.05, 0.025, 0.0166) are considered. For the current configuration, the particle encounters a low Reynolds number (ReD = 612), high Mach number (Ma = 4.1) flow around it. As seen in Figure 11, the bow shock formed upstream of the sphere in both viscous and inviscid cases is nearly identical. However, the flow in the wake region of the spherical particle displays viscous effects. Comparison of the axial velocity contours obtained from the inviscid and the viscous simulations in Figure 12 show that the wake close to the sphere ( XD > 10) is wider for a viscous flow compared to the inviscid flow. However, the width of the wake reduces further away from the sphere in the viscous calculation because of the increased diffusion of momentum from the free-stream. The axial velocity component along the axis of symmetry in the wake region is shown in Figure 13. In Figures 11 and 13, the viscous simulation results also show formation of a small recirculation region behind the spherical particle. Further comparison is made of the axial velocity profile along the radial direction at the location of the zenith of the sphere surface in Figure 14. The axial velocity profile in Figure 14 shows formation of the boundarylayer over the sphere in the viscous case, which absent in the inviscid case. Figure 15 shows the comparison between the drag coefficients obtained from the inviscid and viscous simulations against experimental data (Wegener and Ashkenas 1961). The experimentally obtained dragcoefficient for the sphere is 1.226 while the dragcoefficient obtained from the viscous and inviscid calculations are 1.1 and 0.9, respectively. The inviscid calculations underestimate the drag coefficient in the present low Reynolds number (ReD = 619) flow. The skin-friction drag component contributes to 14% of the total drag of the sphere immersed in the viscous flow. In Figure 16, it is shown that the difference between the computed viscous drag coefficient and the experimental value monotonically decreases with grid refinement. The error in the drag coefficient obtained from the finest mesh employed is 14%. This discrepancy is because of inadequate resolution of the boundary layer over the sphere. Resolution of the boundary layer in 3D on a uniform Cartesian grid is computationally expensive. Local mesh refinement (Berger and Oliger 1984) is expected to provide adequate resolution of the boundary-layer and will be deployed in future work. However, the results show that the current sharp interface method in conjunction with the modified GFM is capable of 3D computations and provides reasonably accurate drag for viscous supersonic flow over spheres.

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Figure . Comparison between the average drag response of the particles in the cloud, obtained from viscous and inviscid simulations of a particle cloud interacting with a shock of strength Mach ..

3.5. Shock-particle cloud interaction The interaction of a shock wave with a cloud of particles is computed in this section. A cluster of 158 aluminium particles, distributed randomly within a square shaped area of dimensions 50 × 50, interacts with a Mach 3.5 shock wave. The non-dimensional diameter of each particle is 1.0. Both viscous and inviscid calculations of the shockparticle cloud interaction are carried out. For the viscous ρ psU D flow simulation, the Reynolds number ( = μps p ) of the flow is 500 with respect to a single particle in the particle cloud. The uniform Cartesian grid used in both inviscid and viscous calculation has a resolution of 20 grid points across particle diameter. The initial configuration of the particle cloud immersed in a quiescent fluid is shown in Figure 17(a). As the shock passes over the particle cloud, interactions of shock waves with the particles give rise to an unsteady flow field. The evolution of the intricate shock structures for the viscous case is seen in Figure 17 via numerical Schlieren images obtained at three different instances (t∗ = 0.0, 0.2, 0.4). In Figure 17(b,c), it is seen that the incident shock-wave deforms and its strength attenuates as it traverses the particle cloud because of the transfer of momentum from the fluid to the particle phase. In the sequence of numerical Schlierens in Figure 17, it is observed that the displacement of the particles in the downstream part of the cloud is less than the particles in the front of the cloud. As the shock passes over the cloud, the particles located at the leading edge begin to equilibrate with the flow even before the shock has reached the

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Figure . Vorticity contour computed at t∗ = . from (a) inviscid and (b) viscous simulation of shock-particle cloud interaction.

downstream end of the cloud. Owing to this, the particles at the front-end start moving before the shock reaches the trailing end of the cloud. This leads to enhanced clustering at the leading edge of the cloud, i.e. the local volume fraction of the particles at the front of the cloud becomes high relative to the rear. Therefore, even within the short period of the shock passage, movement of the particles changes the local solid volume fraction in the cloud. A comparison between the shock structures obtained from the inviscid and viscous simulations is presented in Figure 18(a,b). The shock structures inside the particle cloud are more smeared out in the viscous flow simulation. However, apart from such diffusional effects, the shock structure that emerge from the unsteady flow field from the viscous and inviscid simulations are similar. The average drag of the particles in the particle cloud containing n particles, at a given time, is computed using

Figure . Vorticity contour computed inside a representative volume element in the particle cloud at t∗ = ., obtained from (a) inviscid and (b) viscous simulations of shock-particle cloud interaction.

the following equation: Fx,avg (t) =

n i=1

FX (t − τi ) , n

(56)

where, Fx is the force acting on the particle in x-direction i.e. in the direction of the propagation of the shock wave. τi is the time when the ith particle is impinged upon by an incident shock for the first time. The average dragcoefficient of a particle in the particle cloud is obtained from the following equation: CD,avg (t) =

Fx,avg (t) . 0.5ρps u2ps Dparticle

(57)

The subscript ‘ps’ denotes the post-shock flow parameters. The average drag response of the particles predicted

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by the viscous simulation is comparable to the inviscid flow, as seen in Figure 19. Viscous forces appear to have only a small effect on the drag experienced by the particle in the short time span of shock passage over the particle. Viscous simulations, especially in 3D are significantly more expensive than inviscid calculations. Therefore, inviscid calculations are preferred and may suffice to build closure models for the unsteady momentum exchange in the frozen regime in shocked particulate flows. Despite the relative insensitivity of the drag with respect to viscous effects, it is worth noticing that there are subtle differences between the inviscid and the viscous flow fields in the cloud. Figure 20 shows the vorticity fields in the particle cloud for the inviscid and viscous cases. In the inviscid flow field, the vorticity is generated in the wake of each particle because of the baroclinic instabilities in the wake of the particles. On the other hand, in the viscous case, boundary layers form around the particles and a separated shear layer is released into the wake of the particles. The vorticity from the boundary layers adds on to the baroclinic vorticity which enhances the vorticity in the wake in the viscous case, as seen in Figure 21(a,b). Note that while the vorticity pictures for viscous and inviscid cases show significant differences, the overall drag on a representative particle is only modestly affected for the present case. Therefore, the average drag in particle cloud is dominated by the pressure forces exerted by the incident shock on the surfaces of the particles during the period when the shock wave traverse across the particle cloud.

4. Conclusions This paper develops a level set-based sharp interface method for viscous simulations of shocked particle-laden flows. The sharp interface approach allows for imposition of the no-slip boundary condition on the embedded solid–fluid interface. In the present paper, this is done by using a modified (least-squares-based) ghost fluid treatment that relies on RBCs to construct the ghost field. In the present work, only a single cell thick layer of ghost points is used to impose boundary conditions at the immersed interfaces. A numerical scheme is designed to retain high-order accuracy of convective (second-order) and viscous stress (fourth-order) calculations near the immersed solid–fluid interface. Validation and convergence studies show that the current method is capable of accurately representing the solid–fluid interfaces immersed in viscous supersonic flows. Validation against a similarity solution of compressible boundary layer over flat plate shows that the no-slip boundary condition is applied accurately at

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the solid–fluid interface. Study of steady supersonic flow around a cylinder shows that the current method predicts the correct wall pressure and wall normal shear-stress component variations along the interface. Comparison with benchmark results for the 2D simulation of cylinder lift-off caused by shock impingement provides validation for moving boundary problems. Through 3D computation of a low Reynolds number, steady supersonic flow over a sphere, it is shown that the current method is straightforward to extend to three dimensions as well. However, finer meshes than employed in this work are required to achieve converged results; in particular resolution in the vicinity of the solid surface will need to be high in order to accurately capture viscous stresses and heat transport in the thermal boundary layer for nonadiabatic particle surfaces. 3D computations of viscous effects therefore will benefit from local mesh refinement to resolve boundary layers and shocks. The present work sets the foundation for extending studies of viscous shock-particle cloud interactions in several ways. The meso-scale study of shock interaction with a cloud of particles presented in the current work supports the assumption (Regele et al. 2014; Mehta, Jackson et al. 2016) that overall movement of the cluster of particles within the shock interaction time scale is not significant. However, the local movement of the particles within a large cluster may lead to clustering and collisions among the particles. Particle clustering and inelastic collision among the particles in the cluster within the ‘frozen’ period may influence the net momentum exchange between the fluid and the solid phases. In the current work, collision among the particles is not considered. However, implementation of a collision model has been performed in on-going work to study the effect of clustering and collisions within the particle cloud. The current viscous simulation of shocked particle-laden flow reveals that the effect of viscosity on average drag of the particles is small within the shock interaction time scale. However, as the momentum of the particle phase equilibrates with the fluid phase, relative Reynolds number of the flow with respect to the particles reduces further. Therefore, the effects of viscosity may become more pronounced at the longer time scales. Such long-time frame calculations are being pursued in on-going work. The viscous effects dominate diffusion-based processes, such as, interphase heat transfer and mass transfer in shocked particle-laden flows. Quantification of theses diffusionbased processes from meso-scale simulation will be valuable for macro-scale modelling of shocked particle-laden flows. With the current numerical framework for particle resolved viscous simulation of shocked particle-laden flows, now it is possible to estimate such diffusion dominated quantities. The study of heat transfer between solid

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and fluid phases in shock-particle interaction is also currently underway.

Acknowledgments The authors gratefully acknowledge the financial support by the Air Force Office of Scientific Research (computational mathematics program, program manager: Dr Jean-Luc Cambier) under grant number FA9550-12-1-0115. A grant from AFRL, Eglin AFB (program manager: Dr Martin Schmidt) also partially supported this work.

Disclosure statement No potential conflict of interest was reported by the authors.

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