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

T E

d/δ – 14.2 17.3 14.2

shk

h/δ – 0.4 0.4 0.6

shk

Main parameters of the numerical database. For the UBL case, the Reynolds

Table 4.1. x/δ x/δ

numbers are evaluated at = 69.12 and = 100, corresponding to the trailing-edge

0 0

ramp of CSBL-3 and the position of the inviscid shock impingement, respectively. For

the controlled cases, the parameters are evaluated at the trailing edge of the respective

microramp in the USBLI simulation.

4.2 Boundary layer validation 41

ω

T

µ ,

=

µ T

w w

and thus, the equation to be integrated in this case becomes:

ω

2

∂u u T

τ .

=

∂y ν T

w w U ω

s R /T

U (T )

By defining the transformed mean velocity in the sublayer as = , a

w

0

law of the form is obtained: s

U yu

τ .

=

u ν

τ

s

U

Therefore, the velocity exhibits a linear distribution near the wall, similar to the

incompressible turbulent boundary layer. On the other hand, for the overlap region

between the inner and outer layers, as say in subsection 2.1.5, according to Millikan

+ > y/δ <

[61], the following law of the wall for the overlap region (y 30, 0.3) can be

derived for the incompressible case: 1

+ +

u C,

= ln(y ) +

κ

where a logarithmic behaviour is observed. The above results are in agreement with

µ

Prandtl’s mixing length hypothesis, which assumes that first, is proportional to

t l

the product of density, a turbulence velocity scale, and a mixing length . Second,

m

the turbulence velocity scale is assumed to be proportional to the mixing length

l κy

times the velocity gradient. In this context, = in the logarithmic region.

m

Assuming that this hypothesis also holds for compressible flows, it can be assumed

in the constant stress region that: 2

∂ ũ ∂ ũ

2

2

′ ′

−ρ u ,

u v ρν ρl ρ

= = =

t

g w τ

m

∂y ∂y

where it has been assumed that the Reynolds stresses scale according to Morkovin’s

l κy

approach and that = represents the size of the vortices responsible for the

m

stress. This hypothesis leads to a logarithmic law of variation for the effective

velocity, as defined by Van Driest [75], which takes into account the effect of mean

P r

density variation, assuming = 1, and therefore the recovery factor = 1:

r 1

+ +

u C.

= ln(y ) +

VD κ

This holds in the logarithmic region, but by defining the Van Driest mean velocity

as: u ρ

Z

u du

= (4.6)

V D ρ

0 w

which accounts for compressibility effects through density variations, it is possible

to define a mean Van Driest transformed velocity distribution throughout the entire

domain in the wall-normal direction, in accordance with Guarini et al. [76].

4.3 Microramp wake validation 42

A comparison of the Van-Driest mean velocity profile and velocity statistics with

other data, following the scaling process described above (if necessary), is shown

in Figure 4.2. As a reference case for comparison, the velocity profile from the

uncontrolled simulation (USBL) at the trailing edge of the microramp for CSBL-1 is

Re Re

considered, with = 476.7 and = 1957.7. The results are compared with DNS

τ θ Re

data from Pirozzoli & Bernardini [77] (M = 2, = 583), and incompressible

∞ τ

data from Jiménez et al. [78] (Re = 577.8). A satisfactory agreement of the

τ

data is observed both in the viscous sublayer and the logarithmic region, with

small differences in the outer layer, where the effect of the variation in Reynolds

number across the different simulations becomes noticeable. In Figure 4.2(b), the

density-scaled Reynolds stresses are shown. It is observed that, both near the wall

and in the outer region, the stresses smoothly approach zero. In the first case, this

is because molecular friction stresses dominate near the wall, while in the second

case, it occurs due to the reaching of the boundary layer edge, which corresponds

to the non turbulent freestream flow. Also in this case, reference simulation data

collapse well with the comparison data, and the figure clearly highlights the same

′2

u

trend as the incompressible case. In particular, a peak of normalized is visible

+ ≈

y

(occurring in all cases at 15), with a small difference in the peak intensity,

which increases as the Reynolds number rises.

(a) (b)

(a) Van-Driest transformed mean velocity profile and (b) density-scaled

Figure 4.2.

Reynolds stress components of the incoming boundary layer for the reference simulation

database CSBL-1 at the trailing-edge ramp. Comparison with DNS reference data by

M Re

Pirozzoli & Bernardini [77] (light blue circles, = 2, = 583) and incompressible

∞ τ

Re

data by Jiménez et al. [78] (yellow squares, = 577.8). Dashed black lines in (a)

τ

represent the universal law of the wall.

4.3 Microramp wake validation

The wake generated by the microramp has been extensively discussed in section

1.4, where it is explained that the device produces a pair of primary vortices that

enhance the boundary layer energy by transferring momentum from the outer

region to the low momentum near-wall region. This wake gradually lifts off from

the surface and weakens as it moves downstream. To validate these theoretical

4.3 Microramp wake validation 43

descriptions, which are also supported by the literature, the wake of the microramp

∗

is characterized in its symmetry plane (z = 0) by defining several key quantities,

for the CSBL-1 simulation. One such quantity is the Favre-averaged wake velocity

ũ /u

in the streamwise direction, , which is locally defined as the maximum

∞

wake

difference between the controlled and uncontrolled velocity profiles. This serves as

an indicator of the intensity of the low momentum region that forms downstream

of the microramp, resulting in a peak velocity deficit compared to the uncontrolled

y

case. The position of the velocity deficit in the normal direction, , marks the

wake

location of the wake, which, as expected, rises due to the lift-off effect induced by

upwash. These parameters are shown in Figure 4.3(a), where the velocity profiles in

the symmetry plane are presented for four different streamwise positions downstream

the microramp, normalized by the ramp height (x/h = 5, 10, 15, 20). As observed,

due to the weakening of the wake and the decay of the vortices, the velocity deficit

ũ

relative to the controlled case decreases, leading to an increase in . Additionally,

wake

x/h = 5), a region of

for positions closer to the trailing edge of the microramp (e.g.,

y/h

flow acceleration is seen at higher values of due to an expansion wave from the

ramp’s trailing edge, which dissipates when it meets the shock. Another important

quantity is the peak of the Favre-averaged velocity in the normal direction to the wall,

ṽ /u y

, shown in Figure 4.3(b). Such a peak, whose position is , quantify

∞

max ṽ

max

the mutual interaction of the counter-rotating primary vortices, which tend to lift

off the wake downstream.

(a) (b)

Favre-averaged velocity profiles along the simmetry plane of the microramp:

Figure 4.3.

(a) streamwise velocity and (b) normal-wall velocity.

ũ /U ṽ /U

The velocities and obtained downstream of the microramp for

∞ ∞

max

wake

the CSBL-1 simulation, up to the interaction region, are shown in Figure 4.4(a) and

4.4(b). These results are compared with experimental PIV measurements conducted

by Giepman et al. [79] and Tambe et al. [80] for a similar configuration respectively

4

× ≈

h/δ M Re h/δ

characterized by = 0.35/0.46, = 2, = 6.7 10 and 0.6,

∞

99 θ inc

4

×

M = 2,Re = 2.4 10 . Furthermore, they are compared with a power-law

∞ θ

inc

fitting proposed by Grèbert et al. [81], which is based on the theoretical framework

introduced by Tennekes et al. [82]. This fitting, derived from LES data, corresponds

h/δ M Re

to a configuration with = 0.476, = 2.7, and = 3600 (based on

∞

99 θ

L

the momentum thickness measured 1.5 upstream of the inviscid impingement

sep

4.3 Microramp wake validation 44

b b

− a(x/h) ũ /U a(x/h) ṽ /U

point), and follows the form 1 for and for ,

∞ ∞

max

wake

a > b <

0 and 0. The non linear fitting performed on our data is shown

with − x /h >

with a solid black line, considering only values for which (x ) 10. The

T E

obtained coefficients are compared with those estimated by Grebért et al. [81] in

Table 4.2 and by Giacomo della Posta et al. [83], showing a constant ratio of 2.2

b b

between and for the three cases. Our data match perfectly with those of

v u

Della Posta et al.[83] except for slight differences in the fitting coefficient of the

upwash velocities. The data shown in Figure 4.4(a) are in agreement with the

experimental results of Giepman et al. [79] and Tambe et al. [80], as well as with

the non linear fitting of Grebért et al. [81], particularly in the far wake region

behind the microramp. Small discrepancies between the experimental results and

the current data are observed in the upwash velocity shown in Figure 4.4(b), in

the region closer to the wake, where the higher upwash velocities observed in the

experimental results are likely due to the combined effect of a lower Mach number

and a slightly higher microramp height. In Figure 4.4(c), the black circles indicate

the position of the microramp wake along the streamwise coordinate for the CSBL-1

simulation. This is compared with the experimental data from Giepman et al. [79]

and Tambe et al. [80], which has been also used in the previous analysis of wake

velocity, as well as with DNS numerical data from Della Posta et al. [27] at two

Re Re

different Reynolds numbers: = 500 (red circles) and = 1000 (dark blue

τ τ

circles). The comparison highlights a strong dependence of the wake position on the

Reynolds number. Specifically, the experimental data exhibit a much faster wake lift-

off, attributed to the higher Reynolds number involved and stronger circulation. In

contrast, the present simulation results show excellent agreement with the observation

of Della Posta et al. [27] regarding Reynolds number dependence, as our simulation

lies between the two reported DNS cases at higher and lower Reynolds number.

Case a b a b

u u v v

Current data 1.31 -0.59 3.16 -1.31

Grébert et al [81] 1.60 -0.64 4