Development of an Unstructured CFD Solver for External Aerothermodynamics and Nano/Micro Flows

The utility of the solver is extended to rarefied gas flow regimes as follows. The influence of the new jump condition on temperature and heat flux for various flow situations has been reported in the literature.

Development of an All-Speed Density-Based Navier-Stokes Solver for

This thesis is partly the result of efforts to develop a 3D unstructured mesh density-based solver at the Indian Institute of Technology, Hyderabad, India. The main focus of this work has been to test and improve recently proposed numerical methods for solving the problem set covered by this thesis.

Turbulence Modeling

Later, we separately present an extensive discussion of how we developed a new method to perform near-wall modeling when using the Spalart-Allmaras turbulence model. In this work, we have used the positive unconditional time-implicit integration procedure for the turbulence model equations developed by Mor-Yossef and Levy [ 29 , 30 ].

Automatic Wall Treatment

This latter approach is known as the wall function method, which is commonly used with a high Reynolds number (HRN) type of the turbulence model [33]. This approach allows the desired automatic switching, between a low-Re formulation and the wall function approximation, based on the grid density.

Non-Equilibrium Boundary Conditions

The absence of a velocity contribution causes the result obtained from CFD to differ significantly from the DSMC results for heat transfer. Maslen [64] overcame this problem by adding a sliding friction term to the heat transfer calculation obtained from the sliding velocity boundary condition applied to the surface.

Applications to Micro/Nano Flows

However, they did not investigate the effect of the jump condition on the prediction of other flow properties. There is a need for a more comprehensive analysis of the effect of the Le temperature jump boundary condition on all relevant flow properties for different flow conditions.

Hypersonic Flow over a Cylinder

However, they did not consider the behavior of different convective schemes with respect to the shock anomalies. 12] SLAU2 still showed inconsistencies in the prediction of heat flux for the case of hypersonic flow over a blunt body.

Thesis Outline

The original idea of ​​SLAU2 has been to add dispersion to the numerically captured shock according to the strength of the shock. In this chapter, we perform a comprehensive analysis of a recently developed temperature jump condition and propose a possible correction for the observed anomaly in the results.

Navier-Stokes Equations

2πd2ref(5−2¯ω)(7−2¯ω) (2.11) where ¯ω is the temperature exponent VHS, m is the mass of a gas molecule, dref is the diameter of the molecule, Tref is the reference temperature and kB is the constant of Boltzmann. An integral form of the preconditioned compressible Navier-Stokes equation for an arbitrary control volume Ω with an elementary surface dS is written as [8],.

Figure 2.1: A finite control volume (non-moving).

Spatial Discretization

• Convective Flux
• Viscous Flux
• Solution Reconstruction
• Green-Gauss Method
• Limiters

Therefore, the flux calculations for these currents should ensure that only (or mainly) the cell center values ​​on the upstream side of the face are taken into account in the interpolation. The SLAU2 scheme is an extension of the SLAU ((Simple Low-dissipation AUSM (Advection Upstream Splitting Method))) scheme.

Figure 2.2: Gradient Computation on a tetrahedral grid. The cross denotes the location where the gradients are evaluated in order to compute the viscous fluxes (mid-point of the face ABC).

Temporal Discretization

For rigid systems (e.g. real gas simulation, turbulence models) ∆et becomes very small and the calculations required to reach steady state can be very large.

Qn0 −ΣfJf,nb∆Qnnb =−Rn0 (2.66) where 0, nb refers to the centers of the cell or its neighbors, Jf,0/nb is the Jacobian matrix of the discretized current flow (Fn−Gn) with respect to the variables at 0 or nb cell centers. The discretized current on the f side is a function of the value of the 0th cell as well as the value of the adjacent cell, so the adjacent nbterm appears in the above equation.

Courant number

However, in this work we limited ourselves to a maximum CFL of 500, because there is not much change in the convergence rate at large CFL. The global time step used for the calculation is taken as the minimum of all local time steps in the entire zone.

Initial and Boundary Conditions

• Inviscid wall
• No-Slip Wall
• Non-Equilibrium wall
• Pressure Farfield
• Pressure Inlet
• Symmetry Plane

This requires specification of total pressure (Po), total temperature (To) and the static pressure (Pb) at the inlet. 2.90) We then use the isentropic ratio for total pressure, Eq. Using the temperature we obtain the sonic speed at the inlet and thus the velocity is obtained based on the Mach number.

Closure

These require the governing equations to be Favre (for compressible flows) or Reynolds averages of the Navier-Stokes (for incompressible flows) equations with an appropriate turbulence model. The primary complexity introduced by averaging is that non-linear terms in the governing equations result in open-ended terms that cannot be fully determined by the quantities being calculated.

Governing Differential Equation

Reynolds and Favre averaging

Favre- and Reynolds-Averaged Navier-Stokes Equations

But for compressible flows, it gives additional unknown quantities related to the statistical moments of the density. This leads to the notion that the turbulent shear stress is proportional to the mean stress velocity, as is the case in laminar flow.

Turbulence models

Spalart-Allmaras (SA)

It is robust, converges quickly to the steady state, and requires only moderate grid resolution in the near-wall region. Here Cb1 and κ are two more model constants, d is the distance from the nearest wall, ν =µL/ρ is the molecular kinematic viscosity and S0 is a scalar measure of the vorticity magnitude.

Standard K − model

The inflow boundary condition for the Spalart-Allmaras turbulence model is set to the free-stream turbulent viscosity ratio (µT/µL) obtained from the relevant reference. The uniform free-stream inflow condition is used to set the initial conditions throughout the flow field for the pseudo-transient solution.

Menter’s SST K − ω Model

This modification is done to keep Bradshaw's assumption that the shear stress is proportional to the turbulent kinetic energy k (τ = ρa1K) satisfied in adverse gradient flows where the ratio of production to dissipation can be significantly large [129]. The mixing function F2 is used to recover the turbulent viscosity formulation of µT =ρK/ω for free shear layers (where Bradshaw's assumption does not hold).

Near Wall Physics

The wall function treatment and the low Reynolds number method are two alternative approaches for modeling the near-wall region. On the other hand, the low Reynolds number method uses a very fine grid near the wall to resolve the viscous sublayer.

Time Integration Method

Theoretical Motivation

This can be avoided by using temporal linearization of the negative part of the source term [29]. The details of the proof and derivation are reported in the work of Mor-Yossef and Levy [30].

Constructing the M matrix

Convective-Diffusion and Source Jacobian

The construction of the convective-diffusive Jacobian (N) is simplified by dividing each of the matrices Ciji, Cijj, Ciji, Cijj into their positive and negative parts as shown below using matrix A. The quantities Nii and Nij below are the diagonal and off-diagonal block matrices of rowi of the matrix N, respectively.

Figure 3.2: Flowchart summarizing the construction of M matrix.

Closure

In this chapter we demonstrate how we developed an automatic wall treatment (AWT) for the SA turbulence model to predict the wall shear stress and heat flux. We validated and verified the proposed formulation [135] for four cases involving adiabatic and constant temperature wall boundary conditions, thus validating the wall function modeling for both energy and momentum equations.

Automatic Wall Treatment

Algorithm for implementing the AWT

Solve for (a) the mean flow equations (Eq.3.5) and (b) the SA equation (Eq.3.12) uncoupled from each other, based on the previous time step values ​​of the flow. At the wall, ˜ν and µT based on AWT are obtained as follows:. ρ we obtain shear stress value τw.

Boundary and initial conditions

Results and Discussion

• Case 1: Turbulent flow over a flat plate
• Case 2: Forced Convection Over a Flat Plate
• Case 3: Turbulent flow over a bump in channel
• Case 4: Turbulent flow over 2D NACA 0012 Airfoil

In contrast to the previous two cases here, the y+ variation first decreases and then increases upstream of the bump, then. a) The u-velocity profile at the outlet. Three meshes are studied with different first cell center distance from the wall which in the final solution (see Fig. 4.13) corresponds to different y+ ranges at both the top and bottom walls of the airfoil. a) The y+ plot along the lower surface of the airfoil with a zoomed view to show its variation in the first half of the airfoil.

Figure 4.1: Case 1: Flow domain and conditions for turbulent flow over a flat plate.

Closure

The reason for a gradual monotonic decrease in the y+ value is due to the presence of a more favorable pressure gradient to the flow compared to the top surface. 4.15 (a) shows the graph of the pressure coefficient along the wall of the airfoil and Fig.

Figure 4.15: Case 4: Flow over the NACA 0012 airfoil with Ma = 0.15 with on Mesh-3 with and without AWT.

Direct Simulation Monte Carlo (DSMC)

Interactions of Gas with the Surface

However, if the molecule turns inelastically, i.e., the entrance angle is not equal to the reflection angle and with a change in momentum and energy, then the interaction is called diffuse [151, 152]. We, however, in this work choose the value as obtained from the relevant references for each test case.

Different types of Boundary Conditions

• Maxwell slip boundary condition
• Smoluchowski temperature jump boundary condition
• The Le Temperature Jump boundary Condition
• Pressure Jump Boundary Condition

This is known as a slip condition, where the velocity and temperature of the gas molecules near the surface are not the same as the surface velocity and temperature. In rare conditions, the gas temperature at the surface is not the same as the surface temperature, and the corresponding difference is called a "temperature jump".

Heat transfer in a rarefied gas flow simulations

In this paper, we use the above formulation for the pressure jump condition with Maxwell velocity slip and temperature jump condition Le.

Hypersonic Flat Plate Case

Boundary and initial conditions

Numerical Verification and Validations

Results and Discussion

• Case 1: Metcalf et al.’s, M ∞ = 6.1 and T w = 77K
• Case 2: Metcalf et al.’s, M ∞ = 6.1 and T w = 294K
• Case 3: Becker’s, M ∞ = 12.7 and T w = 292 K
• Convergence Study
• Effect of ζ used for the pressure jump wall boundary condition. 119

Model PC3 also gives a somewhat closer match to the DSMC result for the surface gas temperature compared to PC2. The use of a pressure jump with the temperature jump condition Le is suggested as a possible remedy for pressure over-prediction.

Figure 5.1: Flat plate problem.

General Problem Setup

Boundary and initial conditions

Results and Discussion

Case 1: Pressure driven flow in a Microchannel with 90 ◦ bend 129

The pressure is normalized using the specified pressure at the outlet (ie 66666 Pa) and the x-axis shows the distance along the lower wall of the microchannel normalized by its length. The temperature increases in the recirculation region at the beginning of wall-3 and then gradually decreases along the wall, following an almost similar profile for all the calculations, as seen in Fig.

Figure 6.1: Grid Independence and Convergence study for the case of a pressure driven microchannel.

Case 3: Nanoscale flat plate

6.11(c)) the results are similar for all three cases and very close to the DSMC. In general, PC2 and PC3 results are closer to the DSMC results than PC1, but not by much.

Case 4: NACA 0012 Microairfoil

While at the low end the PC2 and PC3 results are almost identical, at the high end the PC3 results are closer to the DSMC. For the lower surface, all three cases give almost identical results, but for the upper surface PC3 is closer to the DSMC results.

Figure 6.9: Flow profile at x = 80 nm from the plate tip.

Closure

The effect of the sliding friction component is therefore dependent on the mean free path of the gas, shear stress and the velocity of the flow. In the case of the micro-airfoil, this has been shown to improve the prediction of temperature and slip velocity on the upper surface of the airfoil (Fig.6.13 and 6.15).

Figure 6.13: Variation of temperature along the lower and upper wall of the airfoil.

General Problem Setup

Boundary and initial conditions

Modified SLAU2

12] have discussed the effect of different definitions of the interfacial speed of sound (c1/2) given in the literature for the accurate capture of normal shocks. 7.2(a)), while SLAU2 has shown an asymmetry in predicting cylinder nose shock, it is almost absent for M-SLAU2.

Figure 7.1: Flow domain and mesh used for the flow over the half cylinder (R=38.1 mm).

Results and Discussion

Grid Sensitivity study

The M-SLAU2 scheme for the grid spacing of 2×10−7 m gives a solution that is symmetric over the cylinder and has the smallest solution oscillations. We have taken the grid with the first grid spacing of 2×10−7 m for all cases of this work.

Effect of Rarefaction

With the increase of Kn the value of the heat flux decreases, except where the shock instabilities are larger (see figure Kn, the distance to the shock decreases and the shock becomes attached to the surface of the cylinder.

Figure 7.3: Grid independence study using four meshes with different first grid point spacing.

Closure

In addition, the techniques necessary to solve the rarefied gas flow in the slip regime were incorporated into the solver. Here, it was shown that the effect of the sliding friction component on the improvement of the thermodynamic prediction is case dependent.

Scope for Future Work

In a general sense, the results showed that the M-SLAU2 scheme is a robust and useful convective scheme for hypersonic flows in continuum and rarefied regimes. e). Appropriate models should be included for them. f) The non-equilibrium boundary conditions used in this work show limitations and case-dependent behavior.

Validation & Verification

Inviscid flow over airfoil with transonic free-steam condition

• A finite control volume (non-moving)
• Gradient Computation on a tetrahedral grid. The cross denotes the
• Piecewise linear reconstruction
• Turbulence boundary layer u-velocity profile
• Flowchart summarizing the construction of M matrix
• Case 1: Flow domain and conditions for turbulent flow over a flat plate. 82
• Case 1: Turbulent flat plate with an inlet M ∞ = 0.2 with the four
• Case 2: Flow conditions for forced convection over a flat plate
• Case 2: y + variation of the first grid point for the four different meshes
• Case 2: Forced convection over a flat plate with four different mesh
• Case 3: Flow domain for turbulent flow over a bump in a channel
• Case 3: The y + variation of the first grid point for the three different
• Case 3: Surface Pressure Coefficient for three mesh configurations and
• Case 3: Contours of eddy viscosity (normalized by free-stream laminar

Similar to the analysis we performed using three different non-equilibrium boundary conditions (Table 6.2) in Chapter 5 and Chapter 6, we perform the simulations for case B here. However, in the case of B4, heat flux, temperature jump, pressure and sliding velocity all have similar results using PC1, PC2 and PC3.

Figure A.2: Plot for the velocity profile at Re x = 1.059 × 10 6 .