2.1 Computational grid

In the present work, off-body variables and derivatives are of interest. For this reason, a new computational grid was created with improved topology from previous studies and a new grid independence study has been performed on the wake parameters of interest. The computational grid was created using the commercial software Pointwise using a structured blocking technique. The grid produced by Pointwise was then converted to an unstructured grid, which is required by OpenFOAM. Previous experience with developing grids for OpenFOAM has found that the presence of high aspect ratio cells in the region of the viscous separation interaction leads to inconsistent results. In this region strong gradients are produced in both principal directions and therefore cells of relatively low aspect ratio ( ${<}2$ ) are preferred. A picture of the computational domain and the cells in this region is shown in figure 2.

Figure 2.

Computational domain and grid in separation region (showing $0.25\times$ fine grid density).

Figure 3.

Velocity ( $x$ -component – $\boldsymbol{V}\boldsymbol{\cdot }\hat{i} \,$ ) profile at $x=0.5D$ from base.

To establish grid independence, velocity profiles in the wake were examined at four grids of increasing grid density. The most demanding case in terms of grid requirements corresponds to the highest Reynolds number case ( $Re_{\infty ,L}=10^{5}$ ). The effect of spatial resolution on vertical distributions of the $x$ -component velocity at $x/D=0.5$ is shown in figure 3. The maximum difference for centreline reversed flow velocity between the finest and second finest grids is less than 2 %. A similar grid dependence was observed for the other primitive flow variables in the wake ( $p$ , $T$ , $V_{y}$ ). Vorticity, a first derivative, was examined along the same line at $x/D=0.5$ (figure 4). The low Reynolds number case ( $Re=10^{4}$ ) was not simulated on the finest grid. Because of the decreased requirements for grid refinement at this Reynolds number it was found that 5.3 million cells was sufficient. Wall properties were also examined for the low and high Reynolds number case. At the wall, an average grid convergence index (GCI) of ${<}3\,\%$ was found for pressure, wall shear stress and adiabatic wall temperature. A similar set-up and convergence study was used for the simulations of flow over an adiabatic sphere.

Figure 4.

Vorticity ( $\unicode[STIX]{x1D714}_{z}$ ) profile at $x=0.5D$ from base.

2.2 Legacy data

To supplement simulation data produced in the current work, data from previous work (Hinman & Johansen Reference Hinman and Johansen2016a ,Reference Hinman and Johansen c ) by the authors were also used. Previously, simulations of hypersonic flow over adiabatic circular cylinders were performed for a range of Mach ( $6<M_{\infty }<10$ ) and Reynolds numbers ( $8\times 10^{3}<Re_{\infty ,L}<8\times 10^{4}$ ). In these previous works, the focus was on separation location, wall shear stress and pressure. A detailed grid convergence study was performed for these parameters (Hinman & Johansen Reference Hinman and Johansen2016c ). These data were used to facilitate analysis of a modified free-interaction theory at separation.

3.1 Wake Reynolds number analysis

According to Grasso & Pettinelli (Reference Grasso and Pettinelli1995) free-stream Mach and Reynolds numbers (based on a traditional characteristic length) are not useful in describing the wake topology because they do not account the flow features that occur upstream from the wake. Therefore, a wake Reynolds number should be used. Following the work of Weiss (Reference Weiss1966) and Grasso & Pettinelli (Reference Grasso and Pettinelli1995), a semi-empirical relationship between the wake and free-stream Reynolds number is derived here for laminar flow over an adiabatic blunt body. From Weiss (Reference Weiss1966) and Grasso & Pettinelli (Reference Grasso and Pettinelli1995), the characteristic wake Reynolds number should be based on the properties in the recirculation region or along the dividing streamline. For the appropriate wake length scale, the base half-height is used. Thus the preferred wake Reynolds number $Re_{w,h}$ is given by:

The temperature along the dividing streamline ( $T_{DSL}$ ) is calculated assuming a constant energy through the shear layer (and a recovery factor of $\sqrt{Pr}$ ). Assuming a $2/3$ power law for the temperature dependence of viscosity, the relationship between free-stream Reynolds number to wake Reynolds number is:

The ratio $h/L$ is the ratio between the half-base height and the length scale used to calculate free-stream Reynolds number. In the present case, $L$ is equivalent to the cylinder diameter $D$ . Using the definition of Mach number and the ideal gas law, this becomes:

Using known relationships for isentropic flow, normal shocks and adiabatic recovery temperature, the ratios found in (3.3) have been derived:

In (3.4) to (3.7) the ratios $h/L$ , $p_{B}/p_{o,2}$ and $u^{\ast }$ must be approximated empirically. The height of the wake, and therefore $h/L$ , depends on separation location and vehicle geometry. For the present comparison to circular cylinder, the empirical formula for separation location from Hinman & Johansen (Reference Hinman and Johansen2016c ) is used to approximate this ratio as a function of post-shock Reynolds number:

From the works of McCarthy & Kubota (Reference McCarthy and Kubota1964), Dewey (Reference Dewey1965), Park et al. (Reference Park, Gai and Neely2010b ) and Hinman & Johansen (Reference Hinman and Johansen2016c ), it is observed that the base pressure is typically close to $p_{B}/p_{o,2}=0.03\pm 0.01$ for a circular cylinder over a wide range of Mach and Reynolds number. The value of $u^{\ast }$ for the Chapman similarity profile is 0.587. However, (as pointed out by Denison & Baum (Reference Denison and Baum1963)) this value is typically not reached. For the present calculations, a value of 0.3 was used. For geometries other than a circular cylinder, different values should be chosen for these variables ( $h/L$ , $p_{b}/p_{o,2}$ and $u^{\ast }$ ) by empirical estimates or more rigorous mathematical modelling.

While the current work is primarily focused on an adiabatic wall for simplicity, it is useful to derive an expression for vehicles with an isothermal wall. For example, in § 5.3, vehicles with non-adiabatic wall temperatures are analysed. Therefore, a relationship for the wake Reynolds number for non-adiabatic cases is also derived here. To approximate this case, the dividing streamline temperature is approximated as the average between the isoenergetic temperature and the wall temperature. The basis of this approximation is the fact that the $DSL$ originates at the wall (at $T_{w}$ ) and then must be heated to some fraction of the viscous recovery temperature. The following equation gives the approximation for $T_{DSL}/T_{\infty }$ in this case:

Next, actual wake Reynolds numbers calculated from simulation data are compared to the wake Reynolds number calculated from (3.3). To accomplish this, the properties along the dividing streamline in the simulation results are spatially averaged:

Equation (3.3) is plotted in figure 5. The simulation results for a range of Reynolds numbers at approximately Mach 6 and 10 are shown.

Figure 5.

Comparison of equation (3.3) to CFD results for Reynolds number ratio.

The results of (3.3) show qualitative agreement to the CFD data points with an appropriate order of magnitude. The difference between (3.3) and the CFD data is likely due to the empiricism in the equation formulation, as well as modelling in the CFD that is not accounted for. For example, in the simulations, the temperature-dependent gas properties vary through the shock and therefore the post-shock temperature is reduced. More substantial error is observed between (3.3) and CFD results for the adiabatic sphere at low Reynolds numbers. This is likely because the approximated variables ( $p_{B}/p_{o,2}$ and $u^{\ast }$ ) are different for the axisymmetric case and may possess increased Reynolds number dependence.

Equation (3.3) is a general approximation of the wake Reynolds number ratio. Therefore it can be used along with the criteria given by Grasso & Pettinelli (Reference Grasso and Pettinelli1995) to predict the nature of the wake. Lines of constant wake Reynolds number are plotted in figure 6 as a function of the free-stream Mach and Reynolds number. For reference, the coverage of the planar simulations from the present work are shown. The topological regimes (A–D) can be described as follows. Regime A is the regime of diffusion dominated wake flow. In this regime there is no central region of constant vorticity. In the work of Weiss (Reference Weiss1966) this regime was described as ‘Stokes-like’. In wake regime B, the flow is convective, but diffusion continues to dominate over advection. As Reynolds number increases, a region of constant vorticity appears and grows in size. Flow regime C is the regime where a secondary vortex appears in the corner of the wake flow. It is not clear whether the flow in this regime is safely considered to be laminar. In flow regime D, the flow is highly convective and both the primary and secondary wake vortex develop constant vorticity cores of increasing size. Regime D is where the transition to turbulence and unsteadiness in the near wake is expected to occur.

Figure 6.

Relationship of free-stream Mach and Reynolds number to wake Reynolds number.

3.2 Observations of wake diffusion dominance

In the works of both Weiss (Reference Weiss1966) and Grasso & Pettinelli (Reference Grasso and Pettinelli1995) (following Batchelor (Reference Batchelor1956)) the vorticity profiles were an important point of discussion. In the previous sections it was shown that the characteristic wake Reynolds number is several orders of magnitude smaller than the corresponding free-stream value and a semi-empirical expression was derived. To examine this further, the variation of wake topology in the present simulations is examined and compared to the respective Reynolds number regimes from Grasso & Pettinelli (Reference Grasso and Pettinelli1995) and the expression derived in the previous section.

Figure 7.

Vorticity contours at various wake Reynolds numbers (planar wake).

From figure 7, at a wake Reynolds number of 24 ( $M_{\infty }=6$ , $Re_{\infty ,L}=8\times 10^{3}$ ) the wake is clearly diffusion dominated. The role of convection on the vorticity distribution is negligible. The vorticity contours indicate a constant diffusive flux of vorticity between the opposing streams. At $Re_{w,h}=100$ , the convection of vorticity by the recirculating streamlines can start to be seen. However, the importance of diffusion in the wake is still evident. At a wake Reynolds number of 265 ( $M_{\infty }=6$ , $Re_{\infty ,L}=4\times 10^{4}$ ), convection is evident but the constant vorticity core is relatively small. At the wake Reynolds number of 547 ( $M_{\infty }=6$ , $Re_{\infty ,L}=8\times 10^{4}$ ) vorticity can clearly be seen convecting with the motion of the wake recirculation streamlines. At this Reynolds number, a small low strength secondary vortex has formed. These observations agree with the ranges presented by Grasso & Pettinelli (Reference Grasso and Pettinelli1995).

Figure 8.

Regions of compression and expansion at various wake Reynolds numbers (planar wake).

Finite regions of compression and expansion are known to occur in the near wake and have an impact on the wake topology (Grasso & Pettinelli Reference Grasso and Pettinelli1995). These are also known to be a function of wake Reynolds number. In order to visualize this, the expansion and compression regions were extracted by taking the dot product of the local pressure gradient ( $\unicode[STIX]{x1D735}p$ ) with the local streamwise unit vector ( ${\hat{s}}$ ). In figure 8, the finite regions of compression and expansion are highlighted in red and blue, respectively. As the role of diffusion decreases (with increasing wake Reynolds number) these regions become more prominent. For example, at the wake Reynolds number of 24 the dividing streamline of the near wake experiences compression for its entire length. However, both compression and expansion are seen when lower diffusion is present.

Results at approximately Mach 10 are also included in figures 7 and 8. The observations and theoretical considerations presented in this section demonstrate that the state of the wake (diffusive or convective) and the observed topologies corresponding to that state are primarily a function of the pertinent wake Reynolds number. The wake Reynolds number has a nonlinear relationship to the free-stream Mach and Reynolds number. Due to compressibility and the presence of shocks, the characteristic wake Reynolds number is significantly smaller than the free-stream value. In many cases, a high Mach number flow can result in a laminar diffusion dominated wake even when the free-stream Reynolds number is high.

4.1 Control volume analysis of a planar wake

In order to begin analysing the near-wake recirculation feedback mechanism that joins the various interactions depicted in figure 1, a control volume analysis is used. Control volume (CV) analysis of the recirculation region is a simple yet powerful tool to demonstrate some of the observed near-wake relationships in terms of a simple force balance. The centreline, base, and dividing streamline enclose a system with no net mass transfer (and hence no convection) in or out. This means that the only terms appearing in an analysis of the volume are forces. Moreover, even though the dividing streamline and base surface have curvature, they can be approximated by the line segments shown in figure 9.

Figure 9.

Control volume for recirculation region analysis.

The force balances in the $x$ and $y$ directions from figure 9 can be rearranged for the ratio between wake length ( $L_{w}$ ) and wake base height ( $h$ ) yielding:

From (4.1), several conclusions can be made regarding shear stress and pressures in the near wake. For a wake of finite size ( $L_{w}/h$ or $h/L_{w}$ ) to exist, it can be seen from (4.1) that $\unicode[STIX]{x1D70F}_{DSL}\neq 0$ . For $\unicode[STIX]{x1D70F}_{Base}$ (treated as positive in the $-y$ direction) the equation demonstrates that it cannot be equal and opposite to $\unicode[STIX]{x1D70F}_{DSL}$ . If it is assumed that the shear stress terms are always positive in the direction given in figure 9, it can be shown that for the above equation to hold the following inequality must be true:

Given that the pressure is non-uniform along the CV boundaries, it is non-physical to assume a constant pressure recirculation region, which is often done in simple analyses (Chapman Reference Chapman1950; Delery & Lacau Reference Delery and Lacau1988). Regardless of $L_{w}/h$ , the mean centreline pressure must be greater than both the dividing streamline and the base. Additionally, the mean dividing streamline pressure must be greater than the base pressure.

To take this further, it is helpful to approximate the pressure along the each side of the control volume as the average of its endpoints. This is equivalent to assuming that the pressure varies linearly along each streamline. While this is not strictly true, it is useful for discussion and will be scrutinized in the following section. The approximations of the average streamline pressures are:

By combining these approximations with the inequality in (4.2) and simplifying, it can be shown that $p_{B}>p_{S}$ , $p_{R}>p_{B}$ , and $p_{R}>p_{S}$ . These can be combined to produce the following inequality:

This conclusion agrees well with conventional observation of the separation interaction that show that $p_{B}>p_{S}$ (Hinman & Johansen Reference Hinman and Johansen2016c ). Therefore, the highest pressure in the near wake must be at the reattachment point. Because at each streamline end point the static and stagnation pressures are equal (i.e. $p=p_{o}$ ), these observations also lead to the inequality for streamline stagnation pressures:

Additionally, applying the streamline pressure approximations (4.3) to (4.5) directly to the force balance in (4.1) yields:

From this approximation it can be concluded that the base pressure and separation pressure cannot be equal. This is not a surprise because of the well-known lip-shock formation process. A more interesting observation is that the ratio between the length and height of the near wake can be approximately understood as the ratio between pressures and shear stress in the wake. Since $p_{B}-p_{S}$ consists partly of the lip-shock pressure rise, an explicit connection between separation and reattachment shocks via near-wake pressure, shear stress, and size is illustrated. For a given wake size ( $L_{w}/h$ ) a stronger recompression shock (corresponding to a higher $p_{R}-p_{B}$ ) must correlate to a higher shear stress along the dividing streamline. This would also require an increase in lip-shock strength ( $p_{B}-p_{S}$ ). Similarly under a constant shear stress, an increase in $L_{w}/h$ indicates a decrease in lip-shock strength ( $p_{B}-p_{S}$ ) and an increase in reattachment pressure rise ( $p_{R}-p_{B}$ ). These hypotheses agree with conventional observations of wake Reynolds number dependence where $L_{w}/h$ increases with Reynolds number and the pressure rise at reattachment increases, and the pressure rise at the base is decreased.

4.2 Streamline pressures

In the control volume analysis, conclusions and general requirements were presented for the pressure along the wake streamlines. Here, simulation results are used to further discuss these conclusions. The static and stagnation pressures have been extracted from the simulation results at $M_{\infty }=6$ , $Re_{\infty ,L}=10^{4}$ and $Re_{\infty ,L}=10^{5}$ for both an adiabatic sphere and cylinder. The pressures were found by tracing an approximate dividing streamline from the separation point to the reattachment point. Stagnation pressure was calculated by first using the NASA polynomials for total enthalpy to calculate total enthalpy and stagnation temperature. Then stagnation pressure was calculated from the stagnation temperature ratio and the isentropic equations. The pressures are plotted versus a normalized $x$ coordinate in figures 10 and 11.

Figure 10.

Static and stagnation pressures along the near-wake streamlines (a) $Re=10^{5}$ (b) $Re=10^{4}$ (planar wake).

Figure 11.

Static and stagnation pressures along the near-wake streamlines (a) $Re=10^{5}$ (b) $Re=10^{4}$ (axisymmetric wake).

From the results, it is clear that the assumption of a linear pressure variation along the streamlines in the previous section is not valid. The static pressure both increases and decreases along each streamline length. However, the predicted relationships from (4.6) and (4.7) are still observed. At both Reynolds number conditions, the maximum pressure location within the near-wake recirculation region is at the reattachment point. At the lower Reynolds number, the variation of the static pressure along the streamline is greatly reduced. Thus at high Reynolds number diffusion plays a smaller role and the local static pressure is strongly driven by local changes in velocity and Mach number. This agrees with the observed differences between topology regimes A and C described in § 3 where finite regions of compression and expansion were observed as local wake Reynolds number was increased.

Stagnation pressure increases along the length of the dividing streamline. The mechanism of ‘division’ as the dividing streamline approaches the reattachment point is stagnation pressure deficiency of the inner streamlines. Thus, the mechanism by which this dividing stagnation pressure is achieved is crucial. A major assumption in the Chapman (Reference Chapman1950) type of analyses is that the dividing streamline is accelerated up to a similarity value and subsequently decelerated in an isentropic fashion. If this were the case, the results would show a stagnation pressure that increased up to a constant value that remains constant to reattachment. Instead, the results indicate that there is a continuous increase in stagnation pressure and therefore the isentropic relations based on a constant stagnation pressure are not valid. This alone does not indicate that the streamline is not isentropic however. This is further investigated in the wake streamline equation analysis.

Along the centreline the stagnation pressure decreases between the wake reattachment point and the rear-stagnation point. Conversely, the static pressure decreases from reattachment and then increases as the flow stagnates at the base. In the direct vicinity of the base (close to B) the results show that at both high and low Reynolds number a significant variation of pressure exists between the centreline and the dividing streamline. This indicates that an integral or thin-layer-type analysis of this section of the flow will be in error. However, downstream of the immediate aft-body the vertical pressure variation is small for both high and low Reynolds number. This observation indicates that in these regions a one-dimensional or integral approach with a family of wake profiles (such as Crocco & Lees (Reference Crocco and Lees1952) type analysis) is appropriate.

4.3 Wake streamline equation analysis

To analyse the mechanism by which the wake streamlines gain and lose pressure, and thereby link the reattachment and lip shocks, the governing equations are analysed along the enclosing boundaries: the dividing streamline, the wake centreline and the base.

4.4 Rear-stagnation point flow

Another observed mechanism within the near wake begins to occur as the diffusivity in the wake begins to reduce (topology regime B). As the centreline flow approaches the base it undergoes a viscous planar stagnation process. This process creates a peak in pressure above the main base pressure at the rear-stagnation point. At low Reynolds numbers (in topology regime A) this does not occur. As Reynolds number increases this effect becomes more visible. The localized peak in pressure at the rear-stagnation point is visible in the pressure distributions from McCarthy & Kubota (Reference McCarthy and Kubota1964), the previous simulations by Hinman & Johansen (Reference Hinman and Johansen2016c ) and in several capsule studies (Suzuki & Abe Reference Suzuki and Abe1994; Mitcheltree & Gnoffo Reference Mitcheltree and Gnoffo1995; Liever et al. Reference Liever, Habchi, Burnell and Lingard2003; Nakamura et al. Reference Nakamura, Manabe and Nishio2006; Mehta Reference Mehta2006, Reference Mehta2008; McDaniel et al. Reference McDaniel, Wright and Songer2011). The simulation result for pressure along the base of an adiabatic circular cylinder at $M=6$ and $Re=8\times 10^{4}$ is shown in figure 19. The regions of viscous–inviscid interaction are labelled. The separation process begins at approximately $118^{\circ }$ in this case. The stagnation point flow occurs between $180^{\circ }$ and $165^{\circ }$ (approximately 0.26 cylinder radii). Therefore this mechanism occurs over a short distance at the base and will be discussed here as plane stagnation point flow.

Figure 19.

Pressure distribution along the cylinder base (Mach 6, $Re$ 100 000) – planar wake.

Figure 20.

Pressure contours in the planar rear-stagnation point region.

For this analysis, since this mechanism occurs over a short distance and at a relatively low Mach number the flow is analysed as an incompressible stagnation point flow. It is also assumed in the present analysis that the centreline pressure prior to the stagnation interaction is equal to the base pressure after the interaction. This is reinforced by the circular pressure contours visible in figure 20. However, at sufficiently high Reynolds numbers ( $Re_{\unicode[STIX]{x1D707},L}>2.4\times 10^{4}$ based on the present data), the pressure from the stagnation solution can fall below the pressure coming from the free-interaction process. This is slightly visible in the pressure distribution in figure 19.

Figure 21.

Pressure in rear-stagnation point flow (legend in order of increasing $Re_{\unicode[STIX]{x1D707}}$ ) – planar wake.

To analyse the flow, the pressure distribution in the stagnation region is approximated as the Gaussian (following Tu & Wood (Reference Tu and Wood1996)):

where $x_{h}$ is the distance from the stagnation point along the wall and $k$ is a constant. The amplitude of the pressure rise is the difference between the static and stagnation pressure on the centreline ( $p_{o,CL}-p_{CL}$ ). Referring back to figure 10 it can be seen that the stagnation pressure does not change significantly in the direct vicinity of the stagnation interaction. Therefore, using the isentropic definition for the stagnation pressure ratio (4.23) becomes:

Thus by fitting (4.24) to experimental or numerical data, the centreline Mach number $M_{CL}$ , and the constant $k$ can be found. Additional information can be found from the Gaussian fit through a force analysis. The momentum change required to stagnate the jet coming from the centreline must be equal to the integrated pressure along the wall. Therefore:

where $\unicode[STIX]{x1D6FF}$ is the effective width of the ‘jet’ coming from the centreline. Noting that the integral of the Gaussian is simply $\sqrt{\unicode[STIX]{x03C0}}k$ and that $\unicode[STIX]{x1D70C}u_{CL}^{2}=p_{CL}\unicode[STIX]{x1D6FE}M_{CL}^{2}$ this can be rewritten and simplified to:

By the same process, in axisymmetric rear-stagnation point flow:

A small stagnation pressure increase is observed in figure 21, the pressure distributions in the rear stagnation region are plot at a range of Reynolds and Mach numbers. From the figure, the post-shock Reynolds number appears to be the major driver. These distributions were fit using the exponential function in (4.24). With these fits it is possible to estimate the centreline Mach number, and the effective jet diameter of the centreline flow. These are plotted in figure 22. As post-shock Reynolds number is increased, the centreline Mach number increases and the effective jet diameter decreases. These observations are consistent with the increasing convective nature as the wake Reynolds number is increased. As the constant vorticity core grows, this effectively decreases the width of the centreline jet. As well, as the diffusion is decreased the flow in the wake is faster, increasing the centreline Mach number. The same process could be used for an experimental base pressure distribution to estimate centreline Mach number and effective jet width.

Figure 22.

Centreline Mach number predicted from base pressure distribution – planar wake.

4.5 Viscous reattachment flow

In order to understand how the wake recirculation feedback mechanism fits into the overall wake mechanism the reattachment process must be discussed. It is known that reattachment can be adequately modelled by integral momentum viscous–inviscid interaction methods such as those of Crocco & Lees (Reference Crocco and Lees1952) and Reeves & Lees (Reference Reeves and Lees1965). It is also known that free interaction can be used to describe reattachment in boundary layers (Settles & Williams Reference Settles and Williams1982; Delery Reference Delery, Babinsky and Harvey2011). However, wake reattachment requires special treatment for free-interaction theory. Here, a free-interaction theory is presented based on the integral momentum form of the boundary layer equations. The interaction model employed here differs by the equations used in its formulation, as well as the wake specific scaling properties employed. In the present derivation it is assumed that the appropriate scaling properties for the entire wake are the properties at the lip separation shock interaction onset. Therefore, pressure, shear stress, streamwise and stream normal lengths are non-dimensionalized by $q_{e,0}$ , $\unicode[STIX]{x1D70F}_{0}$ , $L_{sep}$ , and $\unicode[STIX]{x1D6FF}_{0}^{\ast }$ , respectively.

5.1 Mechanistic summary

Since the recirculation region forms a set of well-defined boundaries between the rear-stagnation point, the reattachment point and the separation point, the overall near-wake mechanism can be described by combining the mechanisms along each section. The mechanistic sections are illustrated in figure 24, and each sub-mechanism is summarized here. (i) First, the boundary layer on the body approaching the wake separates and forms the lip separation shock. This process can be accurately described by free-interaction theory as was shown by Hinman & Johansen (Reference Hinman and Johansen2016a ,Reference Hinman and Johansen c ). (ii) Between the separation point and the wake reattachment point, momentum and energy are transferred between the inner and outer wake by processes dominated by diffusion. Pressure is gained along this streamline by diffusion of vorticity normal to the streamline and by non-conservative advection. The process along this streamline leads to a decrease in entropy along the streamline length due to heat diffusion away from the streamline. (iii) The pressure at the reattachment point is dictated by a balance between the maximum pressure that can be gained by the dividing streamline through sub-mechanism (ii), and the pressure rise impressed by the reattachment shock wave. The recompression mechanism is a viscous–inviscid interaction that was explained in § 4.5 utilizing an integral formulation of the momentum equation. (iv) Along the centreline, stagnation pressure is lost due to diffusion of vorticity along the centreline and non-conservative advection. In contrast to the dividing streamline, the centreline becomes less dominated by viscous vorticity flux as Reynolds number is increased. At high Reynolds number the contribution from the non-conservative advection of momentum becomes an important means of stagnation pressure loss along the centreline. (v) As the flow approaches the base, a viscous planar stagnation point flow occurs. It was shown that the magnitude of the pressure change in this interaction is primarily a function of centreline Mach number, which in turn is a function of post-shock or wake Reynolds number. The closed loop created by sub-mechanisms (i), (ii), (iv) and (v) is referred to as the recirculation feedback mechanism as it links the separation (i) and reattachment (iii) mechanisms.

Figure 24.

Illustration of overall near-wake mechanism.

The recirculation feedback mechanism can now be demonstrated by linking each of the equations described in the previous sections in terms of pressure. While pressure is chosen here, alternative sets of equations could be derived since all parameters must be continuous around the closed streamlines. Energy, entropy and vorticity are examples of properties that must be continuous when the streamline is closed.

Free interaction as derived by Hinman & Johansen (Reference Hinman and Johansen2016c ) for curved surfaces is:

In this equation $F^{\ast }$ is a correlation parameter between the properties at the interaction onset ( $C_{fo}$ , $M_{e,o}$ , $q_{e,o}$ ) and the pressure rise due to the separation shock ( $\unicode[STIX]{x0394}p$ ). Since this theory is based on the boundary layer approximation, the pressure change is also equivalent to the change in stagnation pressure at the wall: ( $\unicode[STIX]{x0394}p_{o}$ ). Taking the difference between (5.1) at separation and after the lip-shock formation yields:

Here, $\unicode[STIX]{x0394}F^{\ast }$ is the difference between the value of $F^{\ast }$ at separation, and at the plateau value.

The magnitude of the pressure rise in the viscous planar stagnation point flow is given by $p_{o,CL}-p_{CL}$ . This can be rewritten in terms of centreline stagnation pressure and Mach number:

Therefore the total pressure rise between separation and the rear-stagnation point is:

From § 4.3, the relation for the stagnation pressure rise around the streamlines in the wake is repeated here:

Since the changes in pressure around the boundaries must be equal, equations (5.4) and (5.5) must balance. It is useful to normalize (5.4) and (5.5) by the dynamic pressure upstream of the lip-shock interaction. Additionally, in (5.5), the centreline and dividing streamline sections should be separated and the remaining dimensional variables should be normalized in a fashion consistent with free-interaction theory. This process yields:

The $W$ and $A$ terms represent the normalized integrals of viscous vorticity flux and advection, respectively. Equation (5.6) is a non-dimensional representation of the near-wake streamline momentum transfer balance. The mechanism in (5.6) describes the feedback between separation and reattachment shocks. It is useful to examine how reattachment pressure fits into the overall wake mechanism. As well, due to the limitations of assumptions used to formulate $W_{CL}$ (see § 4), it is advantageous to not address this term explicitly. By including the pressure decrease along the centreline explicitly, the mechanism can be written as:

Equation (5.7) and the reattachment process ((4.40) or (4.45) for planar or axisymmetric flow, respectively) must be consistent with each other. These three equations constitute the first overall theoretical framework of the laminar near-wake mechanism including separation, reattachment, rear-stagnation point flow and dividing streamline stagnation pressure gain for both planar and axisymmetric near wakes.

By examining each term, the effect of upstream properties, Reynolds number and Mach number can be discussed. Firstly, it is important to notice that free-stream Mach number does not appear explicitly anywhere in the balance. The primary effect of free-stream Mach number is to alter the post-shock properties. The first term on the left-hand side is the free-interaction formulation from Hinman & Johansen (Reference Hinman and Johansen2016a ). Due to hypersonic similarity (when $M>5$ ), $M_{o}$ is mostly fixed. Therefore the primary coefficient is $\sqrt{C_{fo}}$ , which is a function of post-shock Reynolds number. Thus this term is of order $Re_{\unicode[STIX]{x1D707}}^{-1/4}$ . The second term on the right-hand side is more complicated to discuss. The function $[-\left(1+(\unicode[STIX]{x1D6FE}-1)/2\,M_{CL}^{2}\right)^{-\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}-1}]$ can only vary between 0 and 1, while the term $p_{o,CL}/q_{o}$ is the ratio between the total pressure along the wake centreline and the dynamic pressure at the separation interaction onset. Both of these components of the term are Reynolds number dependent but the overall term should be of order of magnitude 1. The first term on the right-hand side represents the stagnation pressure gained along the length of the dividing streamline. The coefficient is $(\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D714}_{0}/q_{0})\,(L_{sep}/\unicode[STIX]{x1D6FF}_{0})$ . By noting that $\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D714}_{0}/q_{0}=C_{f,0}$ , and that from free-interaction theory $L_{sep}/\unicode[STIX]{x1D6FF}_{0}^{\ast }\sim C_{f,0}^{-1/2}\left(M_{0}^{2}-1\right)^{-1/4}$ it can be shown that the coefficient of this term scales with $\sqrt{C_{f,0}/\left(M_{0}^{2}-1\right)^{1/2}}$ . Thus, the viscous terms on the left- and right-hand side of (5.6) scale together (both with $Re_{\unicode[STIX]{x1D707}}^{-1/4}$ ). The terms $A_{DSL}$ and $A_{CL}$ represent any non-conservative advection along the wake streamlines. Recall that non-conservative advection results from the non-constant density in the advective terms. Normalized by the dynamic pressure at the interaction onset, these terms are of the order of magnitude of 1. All of the normalized terms, $W$ , $A$ , $M_{CL}$ , $p_{0,CL}/q_{0}$ and $\unicode[STIX]{x0394}F^{\ast }$ are still Reynolds number dependent and could be analysed in more detail. For example, it was shown that the free-interaction lip-shock formation could be further decomposed into contributions from boundary layer growth and angle change due to the shoulder expansion (Hinman & Johansen Reference Hinman and Johansen2016a ).

It was previously shown using control volume analysis and pertinent wake mechanisms that the wake size (specifically the recirculation region) should scale such that $h/L_{w}\sim C_{f,0}^{1/2}\left(M_{0}^{2}-1\right)^{1/2}$ . This can also be explained physically using the present theoretical framework. A graphically representation is shown in figure 25. As wake length $L_{w}$ increases, the turning angle at recompression decreases (therefore $p_{N}$ decreases). Additionally, as Reynolds number decreases, the fraction of this pressure reached at reattachment gets smaller. These effects are shown by the multiple dashed lines. The solid lines represent the pressure rise dictated by the recirculation feedback mechanism. At lower Reynolds numbers, diffusion of vorticity across the streamline is increased and therefore stagnation pressure is gained faster. Where the lines intersect (and therefore these two effects are balanced) dictates the wake length and the wake reattachment and base pressure.

Figure 25.

Illustration of balance between recirculation feedback mechanism and viscous reattachment mechanism.

The overall wake mechanism summary has demonstrated that the defining parameters of the near wake are the Mach number ( $M_{0}$ ) and coefficient of friction ( $C_{f,0}$ ) at the shoulder (or at the separation interaction onset/pressure minimum), the post-shock Reynolds number ( $Re_{\unicode[STIX]{x1D707}}$ ) and the wake Reynolds number ( $Re_{w,h}$ ). Based on this, recommendations for how to explain and discuss experimental and numerical observations are given here. For a single geometry, due to hypersonic similarity, $C_{f,0}$ and $M_{0}$ will be primarily dependent on post-shock Reynolds number, $Re_{\unicode[STIX]{x1D707}}$ . Therefore post-shock Reynolds number is the fundamental parameter of interest. When comparing different geometries however, $C_{f,0}$ and $M_{0}$ will be geometry dependent. $C_{f,0}$ is a function of the vorticity generated upstream that can then be diffused into the near wake, and $M_{0}$ is a function of compression and expansion of flow due to geometric features. Therefore, the parameters $M_{0}$ , $C_{f,0}$ and $Re_{\unicode[STIX]{x1D707}}$ are the minimum required when discussing results of different geometries. In all cases, observations regarding topology within the recirculation region should be discussed in the context of wake Reynolds number, $Re_{w,h}$ .

These recommendations agree and in some cases contrast similar work explaining near-wake dependencies and scaling. For example, Grasso & Pettinelli (Reference Grasso and Pettinelli1995) argued that vorticity at the shoulder (or interaction onset) is an important parameter. This agrees with the present work. However, the vorticity is a dimensional parameter and is therefore less useful. More important is the ratio between the vorticity and the dynamic pressure carried by the fluid. Thus the coefficient of friction $C_{f,0}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D714}_{0}/q_{0}$ is the most pertinent parameter. Early on in the development of wake studies it was hypothesized that momentum thickness at separation was the key scaling for distance along the free-shear layer (Denison & Baum Reference Denison and Baum1963; Dewey Reference Dewey1963). In the present work, all distances in the streamwise direction were normalized by the separation length, $L_{sep}$ . This allowed for consistent scaling of all observed mechanisms and successfully led to successful analyses and discussion. These two scaling parameters do not contradict each other. However, momentum thickness would not work as the scaling parameter in the present case where an overall mechanism including separation, recirculation, and reattachment was desired.

5.2 Implications for sting-based experiments

Sting-based experiments are common in the study of hypersonic flows. This is often due to practical considerations based on specific wind tunnel or experimental requirements. Therefore, if the wake region is to be studied it is important to have a fundamental understanding of the impact of the sting. In the present study, it has been shown that the centreline flow plays an important role in several recirculation region mechanisms. First, the wake centreline behaves as a vorticity sink everywhere except in the direct vicinity of the base where the viscous planar stagnation point interaction occurs. The viscous flux of vorticity through the centreline acts to modify the pressure of the streamlines in the wake and therefore the stagnation pressure contained in the wake. When a sting is present, this is not the case. The momentum equation at the wall becomes:

Whether or not the wall is a sink or source of vorticity depends on the pressure gradient impressed on the wake by the outer flow. This is in direct contrast to a streamline, where the flux of vorticity is balanced by both the pressure gradient as well as advection. Referring back to figures 12 to 15, it can be seen that the absence of advection will be most important at high Reynolds numbers and in the axisymmetric wake. In these cases, the pressure gradient and advection are very close to balancing. In the absence of advection, the viscous vorticity flux will be forced to balance the pressure gradient. Instead of primarily behaving as a vorticity sink to the wake flow, from ${\hat{s}}/{\hat{s}}_{R}=0.1$ to $1$ the centreline will act as a vorticity source. At low Reynolds number, where the centreline advection terms are nearly conservative, this effect will still have an impact but the change will be more mild since at low Reynolds number the dominant balance is between pressure gradient and vorticity flux anyway (see figure 16).

As vorticity flux is an important mechanism for momentum transfer in the near wake, the static and stagnation pressures, streamline velocities and wake size should all be affected. This is confirmed by empirical observation of sting based experiments. For example, it was observed in the work by Dogra, Moss & Price (Reference Dogra, Moss and Price1994) that the presence of a sting increased the size of the wake by 26 %. It was also observed that the pressure in the near wake increased. As more vorticity is generated in the wake by the sting, the increased convection in the near wake due to the increased wake Reynolds number will carry this vorticity through the near wake altering the flow. Due to this change in wake behaviour, special consideration should be given for high Reynolds number laminar or transitional flows where the vorticity flux along the centreline will be most drastic. These conclusions demonstrate the value of renewed efforts such as that in the study of Hruschka et al. (Reference Hruschka, O’Byrne and Kleine2011) to provide high quality non-intrusive sting-free wake velocimetry. Another valuable endeavour is the development of free-flight diagnostics such as those performed by Hideyuki et al. (Reference Hideyuki, Tomoyuki, Kazuo, Masathoshi and Katsuhiro2013) and Mudford et al. (Reference Mudford, O’Byrne, Neely, Buttsworth and Balage2015).

The second effect of a sting is that the viscous planar stagnation point interaction will not occur or will be present in a significantly modified form. In addition to the presence of an artificial no-slip wall boundary near the centreline, the finite thickness of the sting will displace the rear-stagnation point location. This is important because high speed reversed flow and/or peak base pressure at the reversed flow impingement has been observed in several recent studies of capsule wake flow (Suzuki & Abe Reference Suzuki and Abe1994; Mitcheltree & Gnoffo Reference Mitcheltree and Gnoffo1995; Liever et al. Reference Liever, Habchi, Burnell and Lingard2003; Mehta Reference Mehta2006, Reference Mehta2008; Nakamura et al. Reference Nakamura, Manabe and Nishio2006; McDaniel et al. Reference McDaniel, Wright and Songer2011).

5.3 Disambiguation of observations in the literature

To demonstrate the value of the theory developed in the current work, a review of hypersonic capsule studies along with added discussion based on the wake feedback mechanism has been performed. Estimations of the wake Reynolds number have been made based on reported values from the studies cited. The estimations are only approximate and in some cases involve gross estimates for parameters such as wall temperature. All calculations were made assuming $h/L=0.5$ and $p_{b}/p_{0,2}=0.03$ .

1. (i) Liever et al. (Reference Liever, Habchi, Burnell and Lingard2003) performed simulations of the compressible Navier–Stokes equations assuming laminar flow for the Beagle II re-entry vehicle. It was reported that as Mach number decreased, separation moved upstream. As well, at low free-stream Mach numbers the wake centreline became supersonic and a noticeable peak in pressure at the rear-stagnation point was observed. Unsteadiness in the wake was observed at Mach 1.5. The Reynolds number was not included in the discussion. These observations are consistent with the present work. At high Mach number, the post-shock and wake Reynolds number of the Beagle were low due to the high change in viscosity across the bow shock (3.3). As the Mach number was decreased the effective Reynolds number of the wake flow was increased, moving the separation point upstream and eventually resulting in unsteadiness. The interesting high centreline Mach number at low free-stream Mach number, similarly was due to the increased effective Reynolds number of the wake.

2. (ii) Utilizing simulations of the compressible Navier–Stokes equations, McDaniel et al. (Reference McDaniel, Wright and Songer2011) studied the wake flow behind the Phoenix re-entry vehicle. The results showed a large secondary wake vortex structure as well as a strong reversed flow along the centreline impinging on the base and resulting in a normal shockwave. This was observed at the peak heating condition for the Phoenix entry. Based on the information given in that work, and the analysis of § 3.1, the observed peak heating condition was approximated to be at a wake Reynolds number of 7500. The high wake Reynolds number explains the supersonic reversed flow along the centreline and the observed importance of the rear-stagnation point interaction in this work. The stagnation point analysis in the present work would not be appropriate for this case as the formulation cannot account for the presence of a shockwave.

3. (iii) Compressible Navier–Stokes simulations were performed by Mehta (Reference Mehta2006, Reference Mehta2008) to study the Apollo, ARD and OREX capsules. The results showed a combination of the observations made by Liever et al. (Reference Liever, Habchi, Burnell and Lingard2003) and McDaniel et al. (Reference McDaniel, Wright and Songer2011). Unsteadiness was observed at low Mach numbers. As Mach number was increased separation moved downstream. No reference was made to Reynolds number in this work. Again, the observations in this work can be attributed to increasing and decreasing post-shock and wake Reynolds numbers. Estimation of wake Reynolds number gives a very high estimate ( $1\times 10^{5}<Re_{w,h}<3\times 10^{5}$ ), indicating why a large secondary vortex was observed.

4. (iv) Mitcheltree & Gnoffo (Reference Mitcheltree and Gnoffo1995) performed simulations of the compressible Navier–Stokes equations assuming laminar flow for the Mars Pathfinder capsule. The results showed a secondary vortex. Estimation from the information given in the paper yields a wake Reynolds number of approximately 6000. This puts the Pathfinder in that study in topological regime D which agrees with presence of the large secondary vortex. As well, the peak pressure on the aft-body was observed at the point of reversed flow impingement on the base. This is due to the occurrence of the plane stagnation point interaction occurring at the base.

5. (v) The MESUR Capsule was simulated by Nakamura et al. (Reference Nakamura, Manabe and Nishio2006) assuming laminar flow. In this work, the separation point moves upstream toward the vehicle shoulder with decreasing Mach number. Reynolds number was not discussed. Similar to other studies, these observations are due to the change in wake and post-shock Reynolds numbers due to free-stream Mach number. No secondary vortex is observed in for this study. Using the method from § 3.1 the maximum wake Reynolds number for this flow is estimated at approximately 200 (topological regime B), agreeing well with the expected behaviour. The peak in base pressure due to the impinging centreline reversed flow became higher as Mach number was decreased. As free-stream Mach number decreased, the wake Reynolds number increased resulting in a higher speed reversed flow and therefore an increased magnitude pressure peak.

6. (vi) In the works of Schrijer & Walpot (Reference Schrijer and Walpot2010), and Walpot et al. (Reference Walpot, Wright, Noeding and Schrijer2012) the base flow of the Apollo capsule was discussed using simulation and experimental data. In the simulations, Reynolds number was varied from approximately $1.75\times 10^{6}$ to $2.88\times 10^{6}$ . Each of these cases were assumed to be laminar flow. In the range examined in these studies, it was observed that Reynolds number had no effect on separation, however for increasing Mach numbers the flow stayed attached for longer. This is due to the low Mach numbers in the cases. In this Mach number range, the bow shock shape and properties across the shock wave are not yet fixed by hypersonic similarity. Therefore in the mechanisms outline in the present study, $M_{0}$ will vary, as well as the pressure gradient and thus the vorticity produced by the fore-body. As well, it can be observed by an approximate calculation that the variation in post-shock Reynolds number was approximately half the variation of the free-stream Reynolds number. This variation, based on the empirical formulation of Hinman & Johansen (Reference Hinman and Johansen2016c ) would only result in a $0.25^{\circ }$ change in separation on a circular cylinder.

7. (vii) The Apollo capsule was also simulated by Sinha & Dey (Reference Sinha and Dey2010). In this work the post-shock Reynolds number was used to explain the change in separation location. This agrees with the present study. Multiple secondary vortices appeared in the wake of this simulation. The wake Reynolds number for this simulation has been calculated at approximately $4\times 10^{4}$ . This high wake Reynolds number (well into topological regime D) explains the appearance of multiple recirculation zones.

8. (viii) Experiments and simulations were used by Suzuki & Abe (Reference Suzuki and Abe1994) to study the flow over a re-entry capsule. In this work, the effect of Reynolds number on the wake were found to be negligible compared to Mach number. The Reynolds numbers were swept from ( $1.4\times 10^{6}$ to $4.2\times 10^{6}$ ) at Mach 3. Using the normal shock equations, this corresponds to a change of post shock Reynolds number of $7\times 10^{4}$ to $2\times 10^{6}$ . Utilizing the empirical relation from Hinman and Johansen (Hinman & Johansen Reference Hinman and Johansen2016a ) this range of Reynolds number would result in a change of less than $0.5^{\circ }$ in separation location. Thus Reynolds number independence is to be expected. In the Mach number range investigated ( $M=2$ to 4) the flow is not yet Mach number independent and therefore the Mach number dependence is also to be expected.

9. (ix) Johansen et al. (Reference Johansen, Danehy, Ashcraft, Bathel, Inman and Jones2013a ) examined the Mars Science Laboratory (MSL) capsule utilizing PLIF imaging. It was observed that the separation angle of the wake agreed qualitatively with the empirical formulation for separation on a circular cylinder based on post-shock Reynolds number. This agrees with the hypotheses made in the present study. Although the prediction was satisfactory, non-spherical shape of the MSL would have an impact on $C_{f,0}$ and $M_{0}$ . Further analysis on the sensitivity of these parameters to vehicle shape is needed.

10. (x) Olynick et al. (Reference Olynick, Chen and Tauber1999) performed compressible Navier–Stokes simulations assuming laminar flow for the Stardust Sample Return Capsule. In this work, alternate fore-body sphere–cone geometries were tested, as well as the effect of vibrational relaxation time. It was observed that as the shock stand-off distance increased, the Mach number around the vehicle shoulder was decreased and therefore pressure and heating were increased in the wake. From (5.6) to (5.7) it is clear that a decreased shoulder Mach number would increase the pressure rise. Additionally, the modified fore-body would result in a different vorticity and dynamic pressure (coefficient of friction) upstream of the near wake.

11. (xi) Using DSMC simulations, the Stardust Sample Return Capsule wake was examined by Zhong et al. (Reference Zhong, Ozawa and Levin2008a ). The sensitivity of the wake to altitude was examined. It was shown that for increasing free-stream density (decreased altitude) the wake recirculation increased in size. This result agrees with the present theory (see figure 25). The increased density in relation to viscosity (and therefore Reynolds number) results in a larger wake.

12. (xii) A generic blunt-body shape was investigated using DSMC by Zhong, Ozawa & Levin (Reference Zhong, Ozawa and Levin2008b ). In this work it was shown that for an increasing free-stream number density the wake length significantly increases. It was also shown that the wake size is directly related to the base radius. Additionally, in this work it is said that Reynolds number and Mach number have little effect on the wake. However, this conflicts with the previous observations. Both number density and base radius are directly tied to Reynolds number. Number density appears in Reynolds number by its relationship to density, and base height appears in Reynolds number by its direct relation to the problem length scale and wake Reynolds number length scale. To demonstrate that Mach number had no effect on the wake, the free-stream velocity was reduced. However, as can be seen by the wake Reynolds number dependence discussed in § 3.1, reducing the velocity alone reduces both Mach number and Reynolds number; thereby nearly cancelling out in terms of the effect on wake Reynolds number. The free-stream Reynolds numbers simulated in this paper range from approximately $Re=3\times 10^{3}$ to $2.3\times 10^{4}$ when number density was varied. In this range, the observed change in wake size is expected due to Reynolds number effects.

The relative success of the comparison of the mechanisms from this work to observed flows over capsules in the literature demonstrate the generality of the conclusions. The primary determiner of the near-wake flow is the post-shock Reynolds number, except at low Mach numbers where free-stream Mach number is important. As well, when comparing between geometries, one must provide context by examining the Mach number upstream of the wake ( $M_{0}$ ), the coefficient of friction upstream of the wake ( $C_{f,0}$ ) and either the post-shock or wake Reynolds number.