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 turbu-

lence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

1

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

1.1 Two-dimensional shock-wave/boundary-layer interaction 2

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 inter-

action

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 in-

teraction

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 wu

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