Medida de la linealidad

state, and the conditions set by Hung and Clauss [36] are (D/δ ∼ 2) for a turbulent layer and (D/δ ∼6) for a laminar flow. The expressions appear to highlight a significant difference between small and large protuberances. It appears that for a small protuberance (h/hu) depends on (K/δ) whilst for a long protuberance (h/hu) is scaled by (D/δ).

2.6

Overview of Previous Computational Work

With a considerable database of experimental results available for turbulent and laminar blunt-fin flowfields, the development of numerical schemes to compute these flowfields has grown steadily throughout the years. Initially all computational methods developed were for solving turbulent boundary layers, this bias is partly based on the fact that at the time there was a greater volume of available data for turbulent cases compared to laminar flows. Reflecting this trend in history in the following section, outlined are three different cases of blunt-fin turbulent interactions studied and, two laminar cases from the last thirty years.

2.6.1 Turbulent Computational Work

One of the earliest successful computations of the blunt-fin interaction was presented by Hung and Buning [33] in 1984, and was achieved by solving the Reynolds averaged Navier- Stokes equations for a turbulent flow. The flow was supersonic (M = 2.95), with a (Re/l

= 6.3×107_{) using a (κ−²) turbulence model over a blunt hemi-cylindrical fin with a (D/δ} = 7) at zero incidence. They compared their computations to the experimental results of Dolling and Bogdonoff [16].

Predictions of the pressure field on the nose of the blunt-fin correlated well with the ex- perimental data, but the peak pressure on the stagnation line (related to the impingement of the supersonic jet) was under predicted. Comparison of the streamwise pressure distri- bution on the plate ahead of the fin and at two outboard stations correlated well within experimental data and all the main features such as the upstream influence, the pressure

2.6 Overview of Previous Computational Work S.J.VITHANA rise due to the separation shock, and the double pressure peak (see Figure 2.16), were well simulated.

Corner Vortex Primary Vortex

Figure 2.19: Horseshoe vortex flowfield, particle paths in the plane of symmetry [34]

Calculated particle paths in the plane of symmetry of the interaction (see Figure 2.19) indicated the presence of a large flattened horseshoe vortex structure well ahead of the fin and another smaller secondary corner vortex close to the fin-plate junction. For the case calculated, the separation shock was very weak and was smeared out over several mesh points, therefore, a sharp triple point at the intersection of the bow shock and separation shock was not observed. A strong Mach number variation in the region behind the bow shock close to the predicted pressure peak was not apparent, and so the computations had failed to predict the presence of the supersonic jet which is characteristic of this type of interaction. However the computations indicated that the size and location of the horseshoe vortices is dominated by the inviscid characteristics of the interaction, in particular the bow shock. As a result they summarised the effect (D) has on the flow. The researchers concluded that agreement between the computed results and measured pressure distribution was very good despite the limited mesh resolution. However, the computational techniques needed refining particularly in the areas of shock capturing and, an improved resolution via increased density of mesh points.

In 1994 Tuttyet al [60] simulated a turbulent flow over a blunt-fin/flat plate configuration, and specifically looked at the heat flux on the flat plate and fin. The flow was hypersonic (M = 6.85), with a (Re/l = 5.4×107_{) and a (K/D= 2.8) at zero incidence. The code used}

2.6 Overview of Previous Computational Work S.J.VITHANA was termed SPIKE, a Navier-Stokes code developed by Fluid Gravity Engineering con- taining a Baldwin-Lomax turbulence model. The computational model was built around a blunt-fin with a diameter of 10mm at zero incidence and sweep. Contour maps of the heating distribution around the fin for both experiment and computation were presented. General agreement on the shape of the features was noted. However, the numerical cal- culations overpredicted the location of the initial rise in heat flux upstream of the fin. This discrepancy was thought to be due to the rather coarse grid used in this preliminary study. Reasonable agreement was also shown for the centerline heating distribution, al- though the computation predicted that, close to the fin leading edge, the heat transfer coefficient peaks at 25 times the undisturbed value at the fin leading edge location. Com- parison of experimental oil flow and heat transfer data with the numerical results indicate that the peak in heat transfer close to the fin is due to the impingement of the small corner vortex on the flat plate, and that the peaks and valleys in heat transfer correspond to the local reattachment and separations of the vortical flow structure.

Since then several turbulent numerical studies have been undertaken in hypersonic blunt- fin interactions. A very detailed study of swept and unswept blunt-fin interactions using a turbulent boundary layer were undertaken in 1998 by Yamamotoet al [68]. The flow was hypersonic (M = 3.92), with a (Re/l = 1.26×107_{) and a (D/δ= 10). The numerical scheme} was a high-resolution finite-difference method based on a fourth order compact MUSCL ,TVD scheme with a modified AUSM and the maximum second-order LU-SGS scheme was employed to simulate unsteady flows associated with shock/shock and shock/turbulent boundary layer interactions.

The results obtained for the unswept blunt-fin showed a cyclic system associated with a system of separation shocks produced by several separation bubbles in the separated region. These produce multiple shock/shock interactions with a resultant mixture of several supersonic jets which combine to influence the characteristics of the separated flow (see Figure 2.20). Finally a flow accelerated by the jets becomes a large-scale supersonic jet. The jet streams into the separation bubbles and disturbs them, which causes the number of bubbles to change according to the strength and direction of the jet. This cyclic system was found to dominate the unsteady flow mechanism of the shock/turbulent

2.6 Overview of Previous Computational Work S.J.VITHANA 2.940 Mach No.: 3.920 0.000 0.980 1.960

Figure 2.20: (left) Mach contours in the plane of symmetry of the flow [68] (right) Streamline vortex core paths around the blunt-fin [68]

boundary layer interaction. The results also appeared to indicate that the locations of the vortex centers were time dependent, but the time-averaged surface locations where the vortices are shed remain approximately constant.

Yamamoto also looked at swept blunt-fins and concluded that sweeping the fin from 0◦−30◦ degrees resulted in a dramatic reduction in the upstream influence, a sharp decline in the final peak pressure levels, and generally reduced pressures throughout the entire interaction region. The streamlines in the symmetry plane confirmed that the horseshoe vortices were progressively weakened and reduced in size with increasing sweep.

2.6.2 Laminar Computational Work

In 1994 Lakshmanan and Tiwari [45] analysed a supersonic laminar boundary layer blunt- fin/flat plate junction through computation. They employed a three-dimensional Navier- Stokes code utilising the MacCormacks time-split, finite volume technique and investigated the effect of Mach number and Reynolds number on the flow. The flow was supersonic (M = 2.36), with a (Re/l = 2.7×106_{) and a (D/δ = 7-30) at zero incidence.}

They reported that increasing the freestream Mach number, decreased the extent of sepa- ration ahead of the fin. This was caused due to the subsequent reduction in the bow shock stand-off distance, which would be expected to reduce the size of the interaction region.

2.6 Overview of Previous Computational Work S.J.VITHANA The parametric study also revealed a dependency of the number of vortices with the flow Mach and unit Reynolds number. Moreover, with an increase in the freestream Mach and unit Reynolds numbers, the primary horseshoe vortex bifurcated into two vortices rotating in the same direction (see Figure 2.21). Increasing the Mach number is also reported to reduce the maximum reverse flow velocity close to the plate and in addition lower the height of the reverse flow region.

Primary Vortex Corner Vortex Primary Vortices Corner Vortex

Figure 2.21: Horseshoe vortex flowfield, particle paths in the plane of symmetry [34] (left)Re/l

= 1.2×106 _{(right)Re/l = 5.0×10}6

The most recent numerical studies to date using laminar boundary layers was done by Houwing et al [32]. However the numerical simulations were used mainly as a supple- mentary tool and for some comparative correlations with experimental results. The flow was hypersonic (M = 6.1 - 6.9), with a (Re/l = 4.5×106) and a (K/D = 2.63) at zero incidence. 3000 T 4000 300 1000 2000 90 mm

2.7 Motivation for Present Study S.J.VITHANA