DNS per il controllo dell'interazione urto/strato-limite: effetto di posizione e dimensione di microrampe per il controllo

Direct numerical simulation of controlled shock-wave/boundary-layer interaction: effects of micro-ramps size and location from the interaction zone

Department of Mechanical and Aerospace Engineering

MS Aeronautical Engineering

Melissa Galante

ID number 1893146

Advisor Co-Advisor

Matteo Bernardini Giacomo Della Posta

Academic Year 2024/2025

“Big whirls have little whirls, that feed on their velocity; and little whirls have lesser whirls, and so on to viscosity.” – Lewis Fry Richardson

iii

Abstract

Shock-wave/boundary-layer interaction (SBLI) occurs in most high speed aerospace applications. It leads to adverse effects such as flow separation, localized pressure and heat peaks, increased aerodynamic drag, and significant flow unsteadiness, which are further intensified in the presence of a turbulent boundary layer. As a result, SBLI is a topic of fundamental importance in aerospace engineering, as its uncontrolled development can compromise structural integrity, reduce vehicle controllability, and decrease propulsion system efficiency.

Flow control techniques can help to mitigate these adverse effects and stabilize the interaction. Among passive control strategies, micro-vortex generators represent an optimal compromise due to their effectiveness in improving flow characteristics. Being smaller than the boundary-layer thickness, they energize the low momentum flow near the wall, preventing or mitigating flow separation without introducing excessive parasitic drag.

This thesis employs Direct Numerical Simulations (DNS) to investigate the effects of a microramp on a SBLI induced by an oblique shock impinging on a turbulent boundary layer. The flow under investigation is supersonic, with a Mach number of 2.28 and a friction Reynolds number of 550 at the inviscid shock impingement location.

The study focuses on the influence of microramp size and position relative to the interaction zone to assess their impact on flow control effectiveness. To this end, four cases are compared: a baseline uncontrolled case and three controlled configurations. Starting from a reference configuration in which the microramp is placed at a given distance from the interaction zone, two additional cases are considered, one where the microramp’s distance from the interaction is increased while keeping its geometry unchanged and another where its size is increased while maintaining a fixed position.

DNS data provide a highly accurate characterization of the flow physics and topology for each case, as well as the evolution of the microramp wake. Furthermore, thanks to the long integration time of the simulations, the study provides detailed characterization of the low frequency unsteadiness of the shock foot, a topic that has been relatively poorly explored in the literature.

The results indicate that the microramp wake induces a spanwise modulation of the separation bubble, leading to the formation of a tornado-like vortex whose core is sensitive to the strength of the vortex impinging on the shock foot. The separation bubble, which is reduced compared to the uncontrolled case, appears to be more sensitive to an increase in microramp size, causing a complete downstream shift of the separation line.

Variations in the distance of the microramp from the interaction zone only have a pronounced effect close to the symmetry plane of the microramp. The effectiveness of flow control is evaluated in terms of added momentum flux, revealing that, despite the increase in aerodynamic drag, bigger ramps enable greater near-wall momentum entrainment, which persists also downstream the interaction.

Additionally, some properties of the wake exhibit strong scalability with the microramp height, suggesting that key flow characteristics can be identified using a limited number of parameters. Spectral analysis reveals a consistent increase of the low frequency peak across the entire span. This increase is particularly pronounced at the symmetry plane when the bigger microramp is used for control, suggesting that the motion of the separation shock remains coherent while being influenced by arch-like vortices. iv

Contents

• 1 Introduction 1
• 1.1 Two-dimensional shock-wave/boundary-layer interaction . . . . . . . 2
• 1.2 Three-dimensional shock-wave/boundary-layer interaction . . . . . . 3
• 1.3 Control devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
• 1.3.1 Active vortex generators . . . . . . . . . . . . . . . . . . . . . 6
• 1.3.2 Passive vortex generators . . . . . . . . . . . . . . . . . . . 6
• 1.4 Topology of three-dimensional supersonic flow over a microramp . . 8
• 1.5 Effects of microramps on SBLI . . . . . . . . . . . . . . . . . . . . . 13
• 1.6 Fundamental applications of shock-wave/turbulent boundary-layer interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
• 1.7 Research aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
• 2 Methodology 21
• 2.1 Turbulence fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 21
• 2.1.1 Statistical tools for turbulence . . . . . . . . . . . . . . . . 22
• 2.1.2 Turbulent averages . . . . . . . . . . . . . . . . . . . . . . . 24
• 2.1.3 Equation of mean flow . . . . . . . . . . . . . . . . . . . . . 25
• 2.1.4 Turbulent scales . . . . . . . . . . . . . . . . . . . . . . . . 26
• 2.1.5 Wall-bounded flows . . . . . . . . . . . . . . . . . . . . . . . 28
• 2.2 DNS overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
• 2.2.1 Spatial resolution and time advancing for homogeneous turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
• 2.2.2 DNS computational cost . . . . . . . . . . . . . . . . . . . . 32
• 2.3 Fully compressible Navier-Stokes equations . . . . . . . . . . . . . . 33
• 2.4 Numerical discretization of the Navier-Stokes equations . . . . . . . 34
• 2.4.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . 34
• 2.4.2 Time integration . . . . . . . . . . . . . . . . . . . . . . . . 35
• 3 Numerical setup 37
• 3.1 Computational domain and microramp geometry . . . . . . . . . . . 37
• 3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 40
• 3.3 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
• 4 Numerical dataset and validation 44
• 4.1 Numerical database . . . . . . . . . . . . . . . . . . . . . . . . . . 44
• 4.2 Boundary layer validation . . . . . . . . . . . . . . . . . . . . . . . 39
• Contents v
• 4.3 Microramp wake validation . . . . . . . . . . . . . . . . . . . . . . 42
• 5 Results 49
• 5.1 Instantaneous visualization: qualitative flow description . . . . . . . 50
• 5.1.1 Instantaneous temperature field . . . . . . . . . . . . . . . . 53
• 5.1.2 Instantaneous streamwise velocity field . . . . . . . . . . . . 55
• 5.1.3 Density field . . . . . . . . . . . . . . . . . . . . . . . . . 56
• 5.2 Near-wall behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 60
• 5.2.1 Skin friction lines and three-dimensional flow topology . . . . 62
• 5.3 Mean flow structure . . . . . . . . . . . . . . . . . . . . . . . . . . 79
• 5.3.1 Favre velocities . . . . . . . . . . . . . . . . . . . . . . . . 79
• 5.3.2 Wake characterization . . . . . . . . . . . . . . . . . . . . 83
• 5.4 Added momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
• 5.4.1 Streamwise analysis . . . . . . . . . . . . . . . . . . . . . 93
• 5.4.2 Spanwise analysis . . . . . . . . . . . . . . . . . . . . . . . 96
• 5.5 Analysis of parasite drag . . . . . . . . . . . . . . . . . . . . . . . . 101
• 5.6 Spectral characterization . . . . . . . . . . . . . . . . . . . . . . . . 103
• 5.6.1 Streamwise spectra . . . . . . . . . . . . . . . . . . . . . . 105
• 6 Conclusions 115
• 6.1 Numerical setup summary . . . . . . . . . . . . . . . . . . . . . . . 115
• 6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
• 6.3 Critical aspect and future research . . . . . . . . . . . . . . . . . . 119
• Bibliography 122

Chapter 1

Introduction

Shock-wave/boundary-layer interaction (SBLI) has become a critical area of research over the past few decades due to the challenges it poses to high speed flight systems ([1],[2],[3],[4]). These interactions can significantly reduce the quality of the flowfield, compromising the performance and structural integrity of aerospace systems and leading to undesirable consequences.

The main effect on the flowfield is that the adverse pressure gradient imposed by the shock modifies the velocity profile, by making it less full (altering the shape factor which increases), thus changing the shape of the boundary layer itself, and generating a strong interaction coupling between the viscous flow (i.e. the boundary layer) and the external inviscid flow.

If the adverse pressure gradient is strong enough, the boundary layer can separate, leading to the formation of recirculation bubbles where the flow reverses direction, generating complex flow pattern between vortices and shock. It’s a phenomenon that can occur both in external and internal flows.

On the external surfaces of aircraft, like wing, SBLI can cause control loss, localized peak heating and pressures, unsteadiness and an increase in both thermal and mechanical loads, because of enhanced production of turbulence. In internal flows like supersonic inlet, it can lead to pressure losses, flow distortion, and, in extreme cases, engine unstart [5].

Specifically, the adverse pressure gradient created by the shock can induce large scale intermittent boundary layer separation [6], leading to fluctuating pressure distributions that disrupt the mass flow entering the engine, affecting its performance. So in certain conditions, the entire flowfield can be significantly influenced by large scale fluctuations, which may be periodic or non-periodic.

Phenomena such as shock-induced separation, periodic unsteadiness, and shock oscillations are commonly observed in transonic airfoils, supersonic intakes, overexpanded nozzles, and rotating machinery. These phenomena can lead to a variety of complications, including instability of the airfoil, intake buzz, high side loads, and altered aeroelastic behaviour of compressor blades, resulting in issues like flutter or divergent motion.

Therefore, controlling these phenomena is crucial. Over time, a range of control strategies and devices have been developed to mitigate the adverse effects of separation and other disturbances caused by these interactions, but their geometries, configurations, and applications are still a topic of research.

However, the fluctuations associated with this phenomenon should not always be seen as a bad thing. There are some applications, such as the mixing of fuel and air in supersonic scramjet combustors, where the main problem is the short residence time of the flow in the chamber, which does not allow the two flows to mix properly and initiate combustion. In these circumstances, the interaction of the mixing layer with the shock increase the turbulent intensity and the thickness of the mixing layers, promoting mixing efficiency [7].

Furthermore in certain cases, the interactions leads to the smearing of the shock, weakening it, and reducing wave drag associated with the shock pattern. In all the cases described above, understanding the precise topology of the flowfield is fundamental, and for these reasons this area of research is more focused on computational fluid dynamics, which in most cases shows a reliable flowfield visualization.

Therefore, for a complete comprehension of the phenomenon, in this chapter the basic results from shock-wave/boundary layer interaction theory are briefly summarize, in addition with a description of the main effect of control devices on the interaction, which has already widely discussed in literature.

1.1 Two-dimensional shock-wave/boundary-layer interaction

According to the boundary layer theory developed by Prandtl in 1904 [8], when a fluid flows over a surface, a thin region forms near the wall where viscous effects become dominant. In this region, the fluid experiences significant deceleration, and there are large transverse velocity gradients due to the frictional forces exerted by the surface and the velocity within the boundary layer decreases as it approaches the wall.

When an adverse pressure gradient is imposed, the fluid particles may experience excessive deceleration, causing them to stagnate or even reverse direction, leading to the formation of a separated flow region. An incident shock wave is a classical example of induced adverse pressure gradient that can readily induce separation.

When a shock impinges on a viscous boundary layer, the pressure rise imposed is smeared across the subsonic region of the flow in the near wall area. This causes a progressive thickening of the boundary layer upstream of the shock, eventually leading to flow separation. In this case, the flowfield structure is quite complex as shown in Figure 1.1(a), and a strong viscous-inviscid interaction must S, be considered.

It is possible to observe a flow separation point from which a streamline develops that encloses the separation bubble and ends at the reattachment R. C point Due to the impact transmitted by the impingement shock, the shear 4 layer between the recirculation bubble and the outer flow, deflects, altering the shape of the reattachment region.

To simplify the visualization of the shock pattern, we can replace the viscous part of the flow with an isobaric region (triangular area in p Figure 1.1(b)) at a constant pressure, which corresponds to the plateau pressure 2 predicted by the free-interaction theory (see reference [9]). This isobaric region acts p p like a wall, and because the pressure is higher than the incoming pressure, 2 1 C it generates a separation shock.

This shock interacts with the incident shock 2 C C C C, and they reflect as shocks and, respectively. Shock impinges on the 1 3 4 4 isobaric region at point I, where it is reflected as an expansion wave. This expansion wave modifies the isobaric boundary of the viscous region, causing it to move until it reaches point R.

At point R, a new deflection occurs, leading to the formation of C a reattachment shock, reasonably weak in nature. Since the stagnation pressure 5 1.2 Three-dimensional shock-wave/boundary-layer interaction 3 C C of the flow passing through the incident shock and reflected shocks differs 1 3 C from that of the flow passing through the separation shock/transmitted shocks, 4 a slip line forms at the intersection point H.

So, for a flow with separation, along the streamwise coordinate, a pressure increase is observed at the separation point, followed by a plateau within the recirculation bubble and a subsequent pressure increase at the reattachment zone.

(a) (b)

Two-dimensional flow pattern of a shock-wave/boundary-layer interaction with Figure 1.1. separation bubble: (a) overall flow organisation and (b) inviscid flow pattern [9].

1.2 Three-dimensional shock-wave/boundary-layer interaction

According to Prandtl’s boundary layer theory [8], a two-dimensional flow experiences separation when the wall shear stress is zero and exhibits a negative gradient. However, the concept of two-dimensional separation is sometimes considered a limiting case, as three-dimensional structures are almost always present even in flows that appear to be two-dimensional.

An example of this is the occurrence of Görtler-like vortices, observed in compression ramp separation experiments by Zheltovodov et al. [10]. These vortices develop due to centrifugal forces, which act to move the flow from the outer part of the boundary layer toward concave surfaces.

Therefore, it is crucial to understand the phenomena that occur in the case of three-dimensional separation in order to fully describe the topology of complex separated flows. The velocity profile of the two-dimensional boundary layer flow described u f y above is given by the streamwise component = (y), where is the distance from u e u the wall and the streamwise velocity at the edge of the boundary layer.

In the e three-dimensional case, however, the velocity vector is no longer confined to a plane, −1 w u β w but it can rotate by an angle = tan, where is the transverse component of the velocity profile. Therefore, the shear stress is no longer a scalar quantity, but a vector field, whose components, under the assumption of a Newtonian fluid, are given by:

∂w ∂u τ µ τ µ = = (1.1) x w z w ∂y ∂y w w −x z

The trajectories defined by this vector field in the plane define the so called skin friction lines, which coincide with the streamlines at the wall. The existence of a 1.2 Three-dimensional shock-wave/boundary-layer interaction 4 third dimension allows the fluid to escape from adverse velocity gradients, which will therefore be much higher in the 2D case compared to the 3D case.

Therefore, the 2D description of separated flow, defined by the existence of a bubble containing closed streamlines circling around a common point, is inadequate for three-dimensional flow. For these reasons, according to Legendre ([11], [12]), it is necessary to reconsider the definition of separation for the three-dimensional case.

He introduced the Critical Points Theory, where particular importance is placed on critical points, those where the skin friction value is locally zero. Based on the behaviour of the skin friction lines near these points, the following (shown in Figure 1.2) are distinguished:

• A point where all skin friction lines pass. Depending on the flow Node: direction, it can either be a reattachment node (if all the skin friction lines diverge from it) or a separation node (if all the skin friction lines converge towards it). At a node, all skin friction lines, except one, share a common tangent, unless it is an isotropic node, in which case all skin friction lines have distinct tangents.
• A point where only two skin friction lines pass through. All Saddle point: other lines avoid this point, adopting a hyperbolic shape.
• A point arou