Increasing the Mach number and blunting the leading edge have been shown to reduce the updraft, resulting in a reduction in the extent of separation. The possibility of turbulent reattachment is also confirmed for the leading edge blunting case with these simulations after comparing the results with reported experimental data.
Introduction
In view of this, the initial part of the chapter describes various terminologies and characteristic features of SWBLI. A fundamental understanding of fluid flow is essential to the design of these vehicles that would experience the hypersonic atmosphere.
Shock wave boundary layer interaction
Shock impingement based SWBLI (I-SWBLI)
Ramp induced SWBLI (R-SWBLI)
Interaction of normal shock and boundary layer
Because this type of interaction creates a subsonic flow region behind the interaction, the downstream flow changes can affect the interaction zone. The presence of a forward step in a supersonic or hypersonic flow field also induces this type of interaction, as shown in Figure 1.4(d).
Flowfield in the presence of SWBLI
Such a disturbance propagates upstream of the compression corner through the subsonic part of the approaching boundary layer. Reconnection of the separated flow takes place at station 'R' downstream of the compression corner.
Shock wave boundary layer interaction terminologies
- Upstream influence
- Separation bubble size
- Separation and Plateau pressure
- Peak Stanton number
Therefore, predicting various features of SWBLI becomes a challenging task in the presence of such a complicated flow field. The peak Stanton number is the non-dimensional value of the maximum heat flux on the slope surface.
Literature review
Experimental studies
16], during their studies also focused on the effect of leading edge blunting and the associated entropy layer on the shock-induced boundary layer interaction. In the presence of SWBLI, an almost tenfold increase in surface heating was observed compared to an undisturbed boundary layer.
Theoretical studies
29] recently performed blunt flat plate experiments to investigate the effect of leading edge bluntness on the surface heating due to the interaction of an impinging shock with the boundary layer. Therefore, subsequent studies were carried out by various researchers using the integral form of the boundary layer equations to explain the interaction.
Numerical studies
A comparison of the numerical data in light of previously reported scaling laws for SWBLI was also summarized. Numerical studies of SWBLI in the presence of leading edge blunting were reported by Neuenhahn and Olivier [59].
Objectives of present research
Provision of leading edge bluntness has been considered for separation control in some of the reported studies [14, 16]. Investigate the leading edge effect on axisymmetric shock wave boundary layer interactions in comparison to its two-dimensional counterpart.
Structure of the Thesis
Normalized (non-dimensional) form of the governing equations
In light of this development, the non-dimensional variables are defined with reference to free flow values. Since the present solver considers the non-dimensional form of the equations, the Prandtl number value is needed at each cell centroid instead of thermal conductivity.
Finite volume method
Theoretical formulation of cell-centered FVM 26 especially suitable for treatment of flow in complex geometries. In cell-centered approach, the flow quantities are stored at the center of the grid cells.
Theoretical formulation of cell centered FVM
Whereas, H⊥ is the total normal flux of a face, which includes viscous and inviscid contributions. The conserved variable vector and the source vector in the equations are derived from the cell-centered quantities.
Spatial discretization
Calculation of convective fluxes
Here in the present study, seven different anti-wind schemes are considered, which fall into one of the following two categories. The second category, flux difference splitting schemes are based on the solution of the local Riemann (shock tube) problem.
Reconstruction and implementation of second order spatial accuracy
Here, r~Landr~Rare the vectors pointing from the cell centroid to the face centroid as shown in figure 2.2. So care must be taken in implementing reconstruction to ensure monotonicity while reconstructing left and right variable values.
Calculation of viscous fluxes
A fine tuning of theK is required to maintain an optimal trade-off between convergence and accuracy of the solution. Although the Venkatakrishanan limiter offers significant improvement in solution accuracy and convergence, the computational cost of computing this limiter function is relatively high.
Implementation of boundary conditions
Inviscid wall (free slip) or symmetry boundary condition
The limitation of this boundary condition is that the velocity touches this boundary and the normal velocity component must be zero. The main advantage of the mirror boundary condition is that it allows the use of an interior scheme at the wall boundary and thus takes into account the direction of the propagating waves.
No slip boundary or viscous wall
Pressure Extrapolation Boundary Condition: This is an alternative way of assigning the inviscid wall boundary condition using the fact that there is no flow normal to the inviscid wall.
Inflow/Outflow boundary conditions
Gas models
Perfect gas model
For the perfect gas model, the Prandtl number of the liquid is considered constant. The present solver considers the viscosity model of Sutherland [75] for the calculation of the dynamic viscosity coefficient under the perfect gas model.
Equilibrium flow model
- Implementation of Tannehil Mugge curve fit
One such well-known tabular approach (Tannehil Muge curve fitting) [JC and PH] is used in this present solver to model equilibrium flows. Here in this technique density and internal energy are taken as two input thermodynamic variables.
Temporal discretization
- Simple explicit Euler time stepping
- Five stage Runge-Kutta scheme
- Implicit time stepping
- Relaxation procedure
- Calculation of time step
Therefore, an explicit five-stage Runge-Kutta scheme has been implemented in the present solver to achieve higher order temporal accuracy. Source term that appears in the governing equations (especially for axisymmetric studies in this thesis) is explicitly treated and included in the explicit residual term (Rn) of equation (2.75).
Summary
The validation and verification of the Euler subpart and the full Navier-Stokes solver follow the gas model comparison study. The presented validation studies clearly demonstrate the accuracy and correctness of the in-house developed solver.
Supersonic vortex flow test case
It is observed that the error decreases with mesh refinement by a factor of about 2 in all cases. It should be noted that the limiter does lead to higher error rates, but does not drastically change the order of the solver's accuracy.
Validation of the high temperature equilibrium flow model
Two different cases are studied to analyze the difference in prediction of the perfect gas model and the equilibrium flow model. The temperature distributions of Mach 4 and Mach 20 cases obtained with two different gas models of the current solver are shown in Figure 3.5 and Figure 3.6, respectively.
Hypersonic flow over hemisphere: an inviscid validation study
Therefore, the shock spacing distances obtained from the steady state results are compared in Figure 3.11. The thus obtained shock shape for Mach number 8.0 flow field is compared with corresponding numerical prediction in Figure 3.12.
Laminar supersonic flow with I-SWBLI
The Mach contours of the present study obtained with this grid are therefore shown in figure 3.18. The comparison of predicted skin friction coefficient (Cf) distribution with reported experimental [19] as well as numerical results [41, 84], given in figure 3.17(a), confirms this argument.
Summary
In the high-speed compressible flow regime, accurate prediction of the flow characteristics and its effect on the aerodynamic coefficients is one of the prominent problems faced by aerodynamicists. Preliminary information on these design data can be obtained using the Euler solver for which significant progress has been made in the development of schemes for the calculation of the inviscid fluxes.
Numerical investigations
- Test case 1: Supersonic flow through a ramp in a channel
- Test case 2: Mach reflection
- Test case 3: Hypersonic flow through SCRAMJET intake
- Test case 4: Hypersonic flow over double ellipse configuration
FIGURE 4.5: Comparison of pressure rise across Mach stem predicted by five different schemes for test case 1 (enlarged view) for test case 1 (enlarged view). This test case deals with the shock reflection as Mach reflection from the solid wall.
Summary
Therefore, the main objective of the current SWBLI studies is to understand the boundary layer separation due to the adverse pressure gradient caused by slope-induced shock waves. Such inclination-induced boundary layer separation can lead to significant degradation of the intended performance of various spacecraft components.
Configurations, test conditions and solver settings
Such initial analysis showed that some of the cases are in separated regions, while others are in attached flow regime. Furthermore, the total temperature of all the above conditions is low enough to consider the flow field as ideal gas flow.
Grid independence studies
Grid independence study for Case A
From the Stanton number distribution analysis, it is observed that the heat flux tends to decrease and reach a minimum value as the laminar boundary layer separates on the flat plate far ahead of the ramp foot. Further reattachment of the flow at the ramp surface, over the strong reattachment, shock terminates the separated region.
Grid independence study for Case B
The heat flux and skin friction coefficient are observed to increase rapidly beyond the reattachment point due to recompression of the boundary layer. This increasing nature of heat flux and skin friction coefficient reach a peak value at a location where the boundary layer thickness reaches a minimum, followed by a gradual decrease due to boundary layer thickening and flow acceleration.
Effect of governing parameters on dynamics of SWBLI
- Effect of ramp angle
- Effect of wall temperature
- Effect of variation of freestream total enthalpy
- Effect of variation of freestream Mach number
- Effect of ramp angle variation (Case C)
- Effect of Leading edge bluntness
Effect of controlling parameters on the dynamics of the SWBLI 93 diffuse V- or U-shaped curve at the same location. Numerical simulations are also performed to understand the effect of wall temperature on SWBLI.
Summary
Suggested modifications to the prediction of upstream impact extent, separation bubble size, and maximum Stanton number are seen to improve the applicability of existing correlations. Few researchers have established such correlations for predicting separation bubble size, upstream impact extent, and peak heat flux.
Review of supplementary correlations
In the present case for laminar flow over flat plate, the Blasius boundary layer solution, corrected for compressibility effect together with Eckert's reference temperature method [55], can be applied to find the skin friction and Stanton number distribution ahead of the interaction zone. In addition to these parameters, the calculation of other boundary layer parameters such as boundary layer thickness, momentum and displacement thicknesses, etc., was performed using Eckert's reference temperature method as mentioned [52].
Analysis of extent of upstream influence
Therefore, the final version of the modified Needham and Stollery correlation for upstream influence is as follows. Therefore, equation (6.9) can be treated as the generalized correlation for predicting the upstream influence.
Analysis of separation bubble size
Analysis of separation bubble size 114 constant (Fui) of 6.15 was obtained for the present data set. Analysis of Separation Bubble Size 115 Another important scaling law for separation bubble size is proposed by Katzer.
Studies for separation point pressure and plateau pressure
The correlation for peak heat flux 121 points corresponding to the experimental studies of Holden and Mosselle [100] and the numerical studies of Grasso and Marini [53] are also consistent with the trend line obtained from the modified Needham correlation.
Correlation for peak heatflux
FIGURE 6.11: Current data points fitted according to Grasso and Marini's correlation for peak Stanton numbers. correlation also fits nearly linear trend for the case of ramp angle variation. The steps followed in the formulation of modified correlation for maximum heat flux are given below.
Summary
This chapter reviews the studies related to R-SWBLI in the presence of leading edge bluntness. The dynamics of the interaction between shock waves and boundary layers completely changes in the presence of such bluntness at the leading edge.
Solver settings
In addition, the control strategy requires a separation size lower than the baseline or reference value corresponding to the case of a sharp leading edge. Thus, there are two significant radii corresponding to the maximum separation size and the separation size equal to the size of the sharp leading edge.
Numerical studies
Studies with sharp leading edge plate for R-SWBLI
These parameters for a flat plate with a sharp leading edge are hereby considered as the reference parameters. Comparison of the numerically obtained surface pressure variation from weak and strong interaction theories [105] with the computationally obtained for the flat plate part of the present configuration is as shown in Figure 7.5.
Studies with blunt leading edge plate for R-SWBLI
Blunt leading edge plate studies for R-SWBLI 134 optimum grid level offering a grid independent solution. FIGURE 7.12: Variations of detachment and reattachment locations for different leading edge radii (BL is boundary layer and HEL is high entropy layer) (BL is boundary layer and HEL is high entropy layer).
Effect of governing parameters on R-SWBLI
- Boundary layer thickness and sonic height
- Skin friction coefficient
- Boundary layer edge Mach number
- Wall to boundary layer edge temperature ratio
- Separation and reattachment pressures
FIGURE 7.13: Variation of separation bubble size for different leading edge radii (BL is boundary layer and HEL is high entropy layer). Thus, the evaluated boundary layer thickness can initially be seen to increase with increasing bluntness of the leading edge.
Prediction methodology for “inversion radius”
The prediction methodology for the “radius of inversion” entropy layer 147 then changes properties both at the edge and within the boundary layer. However, changes in the boundary parameters beyond the inversion radius must be attributed to the growth of the boundary layer within the HEL.
Prediction methodology for “equivalent radius”
Prediction Methodology for "Equivalent Radius" 151 that passed through the weakly curved part of the standing shock. For any radius greater than the "radius of inversion", the entropy layer remains thicker than the boundary layer at all locations on the plate.
Summary
Two critical radii were observed during the previous numerical studies for this interaction. Therefore, numerical simulations are performed to understand the effect of these parameters on the size of the critical radii.
Background
A leading edge radius of magnitude smaller than inversion radius has been shown to provide a counter-effect of widening the separation zone. Therefore, computational investigations need further extension to quantify the change in leading-edge bluntness with varying free-stream and wall conditions.
Solver settings and freestream conditions
Furthermore, it has been noted that a leading edge radius of magnitude above the equivalent radius reduces the separation zone size below the sharp leading edge box value. Therefore, it has been concluded that leading edge bluntness of equivalent radius and above must be considered when implementing passive R-SWBLI control technique.
Results and discussions
Effect of freestream stagnation enthalpy
This difference in maximum temperature for sharp and blunt leading edge boxes is more prominent in the case of high enthalpy flow field. Therefore, complete engulfment of the entropy layer by the boundary layer would occur with slightly higher leading edge bluntness for lower free stream enthalpy cases.
Effect of wall temperature
Such identified detachment and reattachment locations for a few selected guide cases are compared in Figure 8.4. In Figure 8.4 we note that the separation point extends beyond the sharp case value at a leading edge radius of 1.2 mm for all wall temperature cases.
Effect of freestream Mach number
Effect of free stream Mach number 166 Boundary layer profiles of two conditions specified above are analyzed using figure 8.9 for some selected leading edge domains. An almost linear upstream shift of separation point for a given leading edge case with decreasing freestream Mach number is observed for small leading edge domains.
Summary
Solver settings
Results and discussion
Grid independence study
Flowfield around the sharp leading edge 2D and axisymmetric configu-
Flowfield around the blunt leading edge 2D and axisymmetric configu-
Wall property variation for 2D and axisymmetric configurations with
Interaction of entropy layer and boundary layer
Summary
Scope of future work
Important physical features of hypersonic flows
The desire to fly faster and higher has revealed different fluid flow regimes to the fluid dynamics research fraternity. Among these special features, the interaction of the shock wave with the thick boundary layer is of exceptional interest to the hypersonic community.
Inviscid shock generation in the presence of compression corner (picture from
Boundary layer growth and associated velocity vectors (picture from Schlichting
Important types of two dimensional shock wave boundary layer interactions
Schematic diagram representing, the 2D high speed flow over a compression
Here, the incipient separation state was defined as a situation where part of the friction distribution on the surface of the skin becomes exactly zero. According to this revised proposal, the incipient separation should coincide with the first appearance of three inflection points in the surface pressure distribution.
Typical pressure distribution along the wall for ramp based SWBLI
Schematic of the flow field for R-SWBLI with leading edge bluntness is as shown in figure 7.1. This effect dampens the process of boundary layer thickening with increasing leading-edge bluntness.
Representation of FVM approaches a) cell centered scheme b) cell vertex scheme 26
Mirror cell approach
Instead, the Prandtl number must be specified in the current solver frame. Therefore, the equation (2.68) for all the finite volumes can be written in the matrix form as,.
Supersonic vortex flow domain
This exact solution is used as the initial condition for the present analysis, and steady-state solutions are obtained in all networks using the AUSM scheme. FIGURE 3.3: Order of accuracy with and without constraints on quad mesh (solid line) and triangular mesh (dashed line).
Order of accuracy with and without limiters on quadrilateral grid (solid line) and
It is clear from this figure that the thermal boundary layer edge temperature increases with increase in leading edge radius. Effect of control parameters on R-SWBLI with leading edge bluntness 145 for the same stagnation enthalpy is seen to hinder the resistance of the boundary layer to separation.
Physical shape of nozzle considered for equilibrium flow gas model validation . 53
Temperature distribution along the axis of nozzle for Mach number 20 case
The solver results are initially validated by comparing the pressure distribution over the surface of the hemisphere with the Newtonian theory [1] and the modified Newtonian theory [1]. According to Newtonian theory, the distribution of the pressure coefficient (Cp) over the spherical surface can be obtained as:
The shock release distance obtained from the solver has been compared with Billig correlation [83] for shock release distance given by , . Apparently, the distance between this property jump location and the stagnation point can be treated as the shock release or shock distance.
Comparison of density variation along the stagnation streamline
Variation of shock stand-off distance with Mach number
Comparison of shock shape for freestream Mach number 8
The upper inclined wall of the computational domain represents the inviscid surface of the wedge (shock generator). This is clear evidence of the computational efficiency of implicit schemes in steady flow simulations.
Schematic of physical situation of shock impingement caused SWBLI
Meshed computational domain for shock impingement SWBLI study
Therefore, boundary layer must be selected at a specific location to investigate the effect of interaction in the presence of leading edge bluntness. It has already been noted that SWBLI changes in the presence of front bluntness.
Comparison of skin friction distributions of viscous flow test case
Mach contours of incident shock-boundary layer interaction study
In the presence of the supersonic flow and the ramp, an attached shock forms at the compression corner and reflects off the top wall, forming a small Mach stem. A comparison of the seven numerical schemes has been performed for the pressure rise above the Mach stem.
Mach contours of supersonic flow past ramp in a channel
Wall pressure distribution for supersonic flow past ramp in a channel
Comparison of pressure rise across Mach stem predicted by five different schemes
Residual fall comparison for test case 1
Schematic diagram of Mach reflection
Domain and boundary conditions used for Mach reflection test case
Non-dimensional density contours of Mach reflection test case
Comparison of Mach-stem height predicted by different methods for M ∞ =
Geometry of the two dimensional SCRAMJET intake
Residual fall comparison for test case 3
Mach contours using AUSM+ for test case 3
Mach contours using Rusanov scheme for test case 3
Comparison of centreline density variation obtained with AUSM+ and Rusanov
Geometry for hypersonic flow over double ellipse
Enlarged view of mesh used for hypersonic flow over double ellipse
Contours of (a) pressure and (b) Mach number for hypersonic flow over a double
Comparison of surface pressure predicted using AUSM+ with reported numeri-
FIGURE 4.20: Comparison of surface pressures predicted by six different schemes for the hypersonic flow over a double ellipse case.
Comparison of surface pressure predicted by six different schemes for the hy-
It has been reported in the literature that the ramp-induced separation of laminar boundary layer depends on various parameters such as Mach number, Reynolds number, wall temperature, ramp angle, etc. If the deflection angle is higher than the initial separation angle, then boundary layer separation occurs.
Schematic of computational domain
Incipient separation condition analysis
Sample mesh used for the simulation of Case A
Stanton number and wall pressure distribution for Case A
Skin friction distribution for Case A
The plateau pressure region of the surface pressure distribution and negative values in the skin friction distribution are calculated for separation.
Convergence histories of different grid levels
Effect of control parameters on R-SWBLI with leading edge bluntness 138 leading edge radius to the first critical radius. Therefore, initial decrease in magnitude of the skin friction coefficient with increase in leading edge radius can be correlated with thicker boundary layer susceptible to separation.
Grid independence study for a ramp based boundary layer separation (Case B) . 89
Effect of ramp angle on pressure distribution
The pressure distribution for ramp angles 100 and 12.50 shows the same trend unlike ramp angle 150 where the marginal region of constant pressure, upstream of the ramp, indicates the presence of a separation bubble. This distribution is seen to follow the typical V-shaped curve near the ramp for the two lower angle test models, while the heat flux distribution at the ramp angle150 follows a.
Effect of ramp angle on heatflux distribution
Effect of ramp angle on skin friction distribution
It is clear from the present studies that the upward influence increases with increasing wall temperature. The wall heat flux is seen to decrease at all locations for increased wall temperature.
Effect of wall temperature on pressure distribution
Effect of wall temperature on surface heat flux distribution
Effect of wall temperature on skin friction distribution
Effect of freestream enthalpy on pressure distribution
Effect of freestream enthalpy on skin friction distribution
Effect of freestream enthalpy on non-dimensional heatflux distribution
Effect of freestream Mach number on Cp distribution
Effect of freestream Mach number on C f distribution
Surface pressure (Cp) distribution (Case C)
Skin friction coefficient distribution (Case C)
Stanton number distribution (Case C)
In the present study, calculations are performed to reveal the effect of leading bluntness for the same configurations experimentally reported by Coet et al. However, the presence of the bow shock for a blunt leading edge slope creates an entropy gradient normal to the flow behind the shock.
Effect of leading edge bluntness on C f
Effect of leading edge bluntness on Cp
Effect of leading edge bluntness on St
Analysis of magnitude of upstream influence 110 The scaling law proposed by Needham and Stollery [3] correlates the upstream influence with the associated parameters as,. Analysis of Magnitude of Upstream Influence 111 facts support the earlier observation of increase in magnitude of upstream influence with increase in wall temperature and ramp angle for given freestream conditions.
Data points fitted according to Needham and Stollery’s correlation
Data points fitted according to Grasso and Marini’s correlation
Data points fitted according to present modified correlation for extent of up-
Completely different aerodynamics can be noticed with the leading edge radius, even upstream of the ramp. This figure clearly shows the continuous reduction of edge Mach number with increase in leading edge radius.
Present data points fitted according to Needham’s correlation (equation (6.10)) . 117
Present data points fitted according to Katzer’s correlation (equation (6.11))
Present data points fitted according to Katzer’s correlation (equation (6.12))
Present data points fitted according to Davis and Sturtevant’s correlation(equation
Present data points fitted according to modified Needham’s correlation(equation
Data points fitted according to separation point and plateau pressure coefficient
Present data points fitted according to Grasso and Marini’s correlation for peak
Therefore, almost all current data points are closer to the fitted line, confirming the improved predictive power of the current correlation. To ensure this observation, some additional experimental data points are also plotted in the same figure.
Data points fitted according to present correlation for peak Stanton number
The present studies have shown the existence of two critical leading edge blunting radii associated with R-SWBLI. Providing leading edge bluntness is the most widely used separation control technique for R-SWBLI and I-SWBLI.
Schematic of R-SWBLI on a blunted leading edge flat plate-ramp model
Computational domain of sharp leading edge case marked with boundary con-
Surface pressure distribution for the reference case
Skin friction distribution for the reference case
Comparison of wall pressure distribution on the plate portion of the model with
Blunt leading edge plate studies for R-SWBLI 133 that these studies belong to the strong interaction case. Details of the computational domains used for blunt leading edge studies are also given in Table 7.2.
Similar investigations of mesh refinement and mesh independence are performed for all the radii investigated.
Comparison of Mach contours for (a) blunt leading edge case and (b) sharp
It can also be observed that the reduction of the peak heat flux in the ramp part of the model exhibits the same trend as that of the peak pressure with increase in the leading edge radius. Here it can be observed that, the separation point moves upstream with initial increase in leading edge radii, while the reattachment point moves downstream for the initial change in radius.
Comparison of surface pressure distributions for various leading edge radii
Furthermore, separation point and reattachment point can be located from Figure 7.10 as the points where distribution of skin friction coefficient crosses the zero line. The thus obtained separation bubble size for all the considered radii is depicted in figure 7.13.
Comparison of skin friction distributions for various leading edge radii
Comparison of Stanton number distributions for various leading edge radii
This critical radius can be termed the 'equivalent radius' for which the separation bubble size is the same as the reference case of a sharp leading edge plate. FIGURE 7.12: Variation of detachment and reattachment sites for different leading edge radii (BL is boundary layer and HEL is high entropy layer).
Variation of Separation and reattachment locations for various leading edge radii
However, in the case of a leading edge radius of 0.5 mm, the entropy layer is seen to be thicker than the boundary layer up to 0.035 m from the leading edge. Furthermore, at a leading edge radius of 1.2 mm, the entropy layer is significantly thicker than the boundary layer at all locations.
