Impact of a gap on boundary layer transition at Mach 615th International Conference on Fluid Control, Measurements and Visualization 27-30 May, 2019, Naples, Italy Impact of a gap

15th International Conference on Fluid Control, Measurements and Visualization

27-30 May, 2019, Naples, Italy

Impact of a gap on boundary layer transition at Mach 6

Yifei Xue1, Z.J. Wang2, Song Fu1,*

1School of Aerospace Engineering, Tsinghua University, Beijing, China2Department of Aerospace Engineering, University of Kansas, Lawrence, Kansas, USA

*corresponding author: [email protected]

Abstract Laminar-turbulent transition in a hypersonic boundary layer can be influenced by imperfections on the

wall surface. Transition can be delayed or accelerated depending on the type, configuration and location of the

imperfections. Both natural transition and the transition triggered by the imperfections are poorly understood. This

paper studies a Mach 6 transitional flow over a flat plate with a gap, which is a simple geometry of the imperfection.

We investigate the interaction between the gap and forcing waves using a high order implicit large eddy simulation

tool. The forcing waves are obtained by combining a two dimensional instability wave and a pair of sub-harmonic

three dimensional instability waves. First, flow structures including expansion and shock waves near the gap are

analyzed to illustrate the basic flow. Second, we compare the behavior of transition processes of two angles of

attack, AoA = 0◦ and AoA = −7◦. The skin friction coefficient distribution with a gap at AoA = 0◦ is almost thesame as that without a gap, which indicates that the transition process is not disturbed by the gap. In contrast,

a slightly earlier transition is observed with a gap at AoA = −7◦ than without a gap. At AoA = −7◦, a Fourieranalysis shows that new disturbances with broadband frequencies are triggered in the gap, propagate downstream

and influence the amplification of the instability modes.

Keywords: Hypersonic transition, implicit large eddy simulation, gap, high order

1 Introduction

Imperfections or roughness elements may trigger an early transition or late transition in a hypersonic boundary

layer. A two dimensional (2D) gap is one of the simplest geometric imperfections. The influence of a 2D gap

on transition has been well studied under subsonic flow conditions. Many criteria [1] are proposed to predict

the effects of imperfections on transition. A gap is favorable to the substantial downstream TS-growth when

the gap is located after the neutral point. To trigger an early transition, the Reynolds number based on the

characteristic length of the imperfection is always found to be larger than a critical Reynolds number.

In transonic flows, the increase in Mach number can stabilize two and three dimensional modes in an

open-cavity flow according to Sun[2]. However, there are few studies on hypersonic transitional flow due to

imperfections, such as the 2D gap. The free stream turbulence is considered to be composed of slow acoustic

waves, fast acoustic waves, vorticity waves and entropy waves. Two discrete modes originating from the slow

and fast acoustic branches in the eigenvalue spectrum[3], fast discrete mode (mode F) and slow discrete mode

(mode S), are the two major modes in a hypersonic boundary layer. When mode S and mode F have the same

phase velocity, a synchronization occurs. We name the point the synchronization point. The boundary layer

is table before the neutral point (the neutral position of the lower branch of the neutral curve) because the

disturbances are damped outside the neutral curve. An imperfection located before the neutral point does not

trigger an amplification of the disturbance. When the imperfection is located between the neutral point and the

synchronization point, the growth of the disturbance can be accelerated. Mode S can be amplified when a bump

imperfection is placed upstream of the synchronization point[4]. The imperfection has the largest influence

when it is close to the synchronization point[5]. In another study, a 2D imperfection located downstream of

the synchronization point is shown to decrease the Mack second mode[6]. However, only little influence on

instability modes such as mode S[7] can be observed with the analysis of stability theory. Transitional processes

interact with two dimensional imperfections (i.e., oblique wave breakdown or subharmonic wave breakdown)

in a hypersonic boundary also need to be investigated.

Besides the existing instability modes, frequency change[8] or new instabilities can also be triggered by

imperfections. For example, the Kelvin-Helmholtz instability of a shear layer[9]. In addition, two kinds of

oscillation mechanisms[10] within the gap are observed in supersonic transitions. Gap flows in this work can

Paper ID:312 1



be classified open cavity flow which has two oscillation mechanisms[11]. The first one is the longitudinal

oscillation which corresponds to the Rossiter mode[12]. The reflected acoustic wave propagates from the aft

corner to the front corner. The other one is the transverse oscillation. The transverse mechanism has a feedback

loop within the cavity. The reflected acoustic wave firstly propagates down to the bottom of the gap and

then transverses to the shear flow. Phenomenons of longitudinal and transverse mechanisms can be observed in

experiments at Mach 1.71[10]. In a hypersonic flow, these new instability modes need to be further investigated.

The main contributions of this study include: the wall-resolved large eddy simulations (LES) of a Mach 6

hypersonic subharmonic transitional flow over a flat plate with a single 2D gap at two angles of attacks, and

a detailed analysis of the forcing instability waves on the accelerated transition process. The neutral positions

of the forcing waves are analyzed with a linear stability theory. The oscillation mechanisms of two angles of

attack are studied by decomposing the instability waves with a Fourier transformation.

2 Numerical Method

This section briefly reviews the numerical method used in the present study. The high order solver, hpMusic, is

based on the flux reconstruction (FR) or correction procedure via reconstruction (CPR) method. This method is

originally developed by Huynh [13] for hyperbolic conservation laws, and later extended to mixed meshes by

Wang and Gao, and Haga et al[14][15][16]. Other developments in the FR/CPR methods are reviewed in [17].

We choose the FR/CPR method because of its ability in handling unstructured meshes, its high-order accuracy,

its simplicity like a finite difference method, and its scalability on supercomputers. The unsteady compressible

Navier-Stocks equations are discretized using the with FR/CPR method. We use the following conservation

law to introduce the basic idea

∂U

∂ t+∇ ·F(U) = 0, (1)

where U is the vector of conservative variables, and F is the flux vector. The computational domain is dis-

cretized with non-overlapping elements Vi. In each element, the conservation law is transformed into a weighted

residual form with an arbitrary test function W

∫

Vi

(∂U

∂ t+∇ ·F(U))WdV = 0. (2)

The conservative variables U is assumed to be a polynomial of degree k, Ui ∈ Pk. Using integration by parts in

the second term, we arrive at

∫

Vi

∂Ui∂ t

WdV +∫

∂ViWF(Ui) ·ndS−

∫

Vi

∇W ·F(Ui)dV = 0. (3)

We replace the normal flux term at element interfaces with a common Riemann flux Fncom to achieve conserva-

tion. Applying integrating by parts again to∫

Vi∇W ·F(Ui)dV , we obtain

∫

Vi

∂Ui∂ t

WdV +∫

∂ViW [Fncom −F

n(Ui)]dS−∫

Vi

W∇ ·F(Ui)dV = 0. (4)

The Riemann flux Fncom is computed with the Roe Riemann solver by the use of Vi and Vi+, in which the subscript

"i+" denotes the neighbor element

Fncom = Fn

com(Ui,Ui+,n). (5)

In order to simplify (4), we replace the surface integral with a volume integral via a lifting operator, δi ∈ Pk(Vi):

∫

∂ViW [Fncom −F

n(Ui)]dS =∫

Vi

WδidV. (6)

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Fig. 1 Schematic of the computational domain

Substituting (6) into (4), we obtain

∫

Vi

[∂Ui∂ t

+∇ ·F(Ui)+δi]WdV = 0. (7)

The aim of introducing the lifting operator is to derive the differential equation from the integral one. As-

suming that the flux vector F(Ui) can be approximated with a polynomial, then we obtain the final formulation:

∂Ui, j∂ t

+∏ j(∇ ·F(Ui))+δi, j = 0, (8)

where ∏ j denotes a projection to the polynomial space and the subscript j represents the projection at the

solution point j, ∏ j(∇ ·F(Ui)) ∈ Pk.

The FR/CPR method converts the weighted residual form from an integral one to a differential one. It is

compact in that the scheme only needs the immediate face neighbors. The viscous flux in the compressible NS

equations is computed with the Bassi-Rebay 2 (BR2) scheme[18]. No explicit subfilter scale models are used.

Therefore the present approach is called an implicit LES or ILES. The backward-difference formula (BDF2)

with a LU-SGS solver[19] and a 3rd Runge-Kutta method are employed for time marching. An accuracy pre-

serving limiter [20] is adopted to capture shock-waves and maintain the high order accuracy of the simulations

elsewhere. The LES tool has gone through an extensive verification and validation process[21].

3 Simulation results and discussions

3.1 Computational setup

To investigate whether a 2D gap affects the subharmonic transition of a Mach 6 hypersonic flow, we perform a

wall-resolved ILES of the transitional flow. In order to avoid the difficulty of simulating a strong leading edge

shock with a high order method, the computational domain starts behind the leading edge and ends after the

formation of the turbulent boundary layer. The gap configuration and the computational domain are illustrated

in Fig. 1.

A laminar boundary layer profile is prescribed at the inflow boundary. The freestream Mach number is

M∞ = 6 and the free-stream unit Reynolds number is Re∞ = 1.0×107/m, the static temperature is T∞ = 55K.

The inflow conditions are of the same order of magnitude as the parameters of the turbulent wind tunnel at

Peking University. The computational domain is 0.02 ≤ x ≤ 0.7m in the streamwise direction and 0.0 ≤ z ≤0.018m in the spanwise direction. Two complete oblique forcing waves are generated at the disturbance stripin the spanwise direction. The instability waves are introduced to the flow by blowing and suction at the

disturbance strip. The gap starts at x = 0.2m and ends at x = 0.22m with a depth h = 0.01m. A periodicboundary condition is used in the spanwise direction. Buffer region are adopted at both the right end and the

top of the domain.

Next we explain the method of blowing and suction. Simultaneous blowing and suction is used in this work

to make sure no additional fluid is added into the flow. The temperature on the disturbance strip is replaced

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Fig. 2 Slice of mesh for AoA = 0◦ gap flow

with an isothermal condition[22]. The temperature equals the adiabatic wall temperature of the steady solution.

Following the blowing and suction function of Huai et al[23] , the wall-normal velocity V is prescribed as:

v(x,z, t) = A2D f (x)sin(ω2Dt)+A3Dg(z) f (x)sin(ω3Dt), (9)

where, A2D and ω2D are the two dimensional wave’s amplitude and frequency, respectively. A3D and ω3D are

the oblique wave’s amplitude and frequency. Then the shape function is defined following Fasel et at[24]:

| f (ξ )|= 15.1875ξ 5 −35.4375ξ 4 +20.25ξ 3,

g(z) = cos(2πz/λz).(10)

ξ can be obtained with xm = (x1 + x2)/2:

ξ =

x− x1xm − x1

x1 ≤ x ≤ xm,

x2 − x

x2 − xmxm ≤ x ≤ x2.

(11)

The location of disturbance strip in the streamwise direction is 0.035 ≤ x ≤ 0.055m (x1 = 0.035m and x2 =0.055m). To obtain a subharmonic transition in the computational domain, a two dimensional wave and a pairof subharmonic oblique waves are added in the disturbance strip. Frequencies of the oblique waves are one half

of that of the two dimensional wave. The resonance of the three waves leads to the subharmonic transition[25].

The disturbance amplitude A and dimensionless frequencies F of AoA = 0◦ are set to:

A2D = 2%U∞, F2D = 0.6×10−4;

A3D = 2%U∞, F3D = 0.3×10−4.

(12)

A negative angle of attack leads to adverse pressure gradient which contributes to the transition. So the distur-

bance amplitudes at AoA =−7◦ are set to smaller values:

A2D = 1%U∞, F2D = 0.6×10−4;

A3D = 0.05%U∞, F3D = 0.3×10−4.

(13)

The ω3D and ω2D can be calculated with:

ω =FU2∞

ν. (14)

The mesh used in the simulation ( i.e., a slice mesh of gap flow at z = 0.009m, AoA = 0◦) is illustrated inFig. 2. Mesh in region x ∈ (0.5,0.7)m is progressively coarsened in the streamwise direction which can reducethe disturbances from outflow boundary. The degree of freedoms (DOFs) in each directions are lists in Table 1

where DOFsin and DOFsout denote the DOFs in the gap and out of the gap, respectively.

3.2 Stability analysis of the instability waves

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Table 1 Grid resolution of the simulation

nx ny nz ∆x+max ∆y

+max ∆z

+max

gap DoFsin 70 81 140

DoFsout 1544 94 140 5.7(AoA = 0◦) 2.5(AoA = 0◦) 2.4(AoA = 0◦)

/10.3(AoA =−7◦) /4.7(AoA =−7◦) /4.3(AoA =−7◦)

smooth DoFs 1544 94 140 5.7(AoA = 0◦) 2.5(AoA = 0◦) 2.4(AoA = 0◦)/10.3(AoA =−7◦) /4.7(AoA =−7◦) /4.3(AoA =−7◦)

Before the implicit large eddy simulation, we need to choose a reasonable inflow disturbance for the hyper-

sonic boundary layer. Stability characteristics of disturbance strip are investigated by LST in this part. We can

compare the gap location with the neutral point and the synchronization point through the stability analysis.

The LST analysis explains why we choose the frequencies of two dimensional wave F2D = 0.6×10−4 and

the subharmonic oblique waves F3D = 0.3×10−4 at the disturbance strip. Here we start with the introduction

of LST[26]. The basic flow for the LST analysis is computed from the compressible Blasius boundary layer

equations. The governing equations derived from compressible Navier-Stocks equations can be separated into

the basic-state equations and the disturbance equations. The form of the disturbances are assumed to be:

q̃(x,y,z, t) = q̂(y)exp(i(αx+β z−ωt))+ c.c., (15)

where c.c. stands for complex conjugate. In spatial stability, α = αr + iαi is the complex streamwise wavenum-ber, β is the spanwise wave number and ω is the angular frequency. We denote the Reynolds number based on

boundary layer thickness with R, and then the eigenvalue problem can be expressed as:

α = f (β ,ω,R). (16)

For β = 0, the neutral curve obtained from LST is shown in Fig. 3(a). Instability wave with a frequency ofF2D = 0.6× 10

−4 will grow within the computational domain. The synchronization point of two dimensional

instability wave predicted in Fig. 3(b) is x = 0.3635m. The gap is upstream of the synchronization point.Growth of instability wave F2D = 0.6× 10

−4 should be affected by the gap according to LST analysis. Gap’s

effect on transition process is discussed with ILES results in the next part.

(a) Neutral curve for β = 0

(b) Phase velocity of mode F and mode S

Fig. 3 LST results

Then we fix R = 741.63 (x = 0.055m, end of the disturbance strip), the contour levels of αi can be obtainedwith LST. As shown in Fig. 4, the oblique disturbance (F = 0.3×10−4, β = 0.0405) locates inside the neutralcurve (αi < 0). So the oblique waves will grow in downstream direction.

We can conclude that the current instability waves will grow monotonously. The oblique transition process

with current forcing frequencies and wave numbers is expected to be affected by the gap.

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Fig. 4 Neutral curve for R = 741.63

Fig. 5 Overall vortex evolution of the gap flow transition process, AoA = 0◦

3.3 Transition processes at two angles of attack

Gap flow of AoA = 0◦

Complete transition processes of smooth and gap flows are obtained with ILES. Similarity results of compress-

ible boundary layer equations are imposed at the inflow boundary. Subharmonic transitions are induced by the

forcing instability waves at disturbance strip. An overview of gap flow transition process at AoA = 0◦ demon-strated with vortices is shown in Fig. 5. Vortices are visualized with Q-criterion colored with non-dimensional

streamwise velocity. The background is the numerical schlieren. Forcing disturbances grow in boundary layer

and trigger the transition. Two pairs of λ vortices are obtained in spanwise direction at the transition onset. The

vortex heads grow in normal direction and interact with the boundary layer when approaching the boundary

layer, as shown in Fig 6(a). Multiple hairpin vortices are observed in Fig 6(b). The hairpin vortices are hard

to be distinguished because the interaction between the boundary layer and the hairpin vortices. Complicated

vortices are observed in the later stage of transition, shown in Fig 6(c). Mixture of large and small vortices

leads to the turbulent boundary layer.

(a) First stage (b) Second stage (c) Third stage

Fig. 6 Three vortex stages of the gap flow transition process, AoA = 0◦

The transition process of smooth flat plate is almost the same with that on gap case. The gap has little effect

on the transition process at AoA = 0◦. However, there are some local flow structures occur at the corner of thegap, as shown in Fig. 5. So the flow at the gap is particularly discussed. Averaged pressure and streamlines are

used to illustrate the flow near the gap in Fig. 7. Three separations are observed in the gap including one main

separation and two small separations on each side of the main separation. Velocity in the gap is mostly less than

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Fig. 7 Averaged pressure distribution and streamwise in the gap at the center z-plane

Fig. 8 Instantaneous slice of streamwise velocity at x-z-plane

1% of the velocity outside the boundary layer which indicates that the gap is nearly a dead zone. The maximum

velocity locates at the upper border of the main separation as a result of the acceleration of the expansion from

the upstream corner. An expansion is followed by a shock at the downstream corner which decelerates the

flow to an undisturbed condition. The expansion and shock are weak that the streamlines at y = 0m are nearlyundisturbed by the gap.

The instantaneous streamwise velocity evolution above the cavity is illustrated in Fig. 8 at the position of

y = 0.0075x+ 0.00105m. A local acceleration is observed at the gap and the streamwise velocity decreaserapidly after the gap. This confirms the velocity acceleration at the gap in the time-averaged results. No special

change on the streamwise velocity is observed far behind the gap comparing with the velocity evolution in

smooth flow.

For the current forcing waves, transition phenomenons of smooth flow and gap flow are the same. The in-

fluence of the gap is limited to the vicinity of the gap based on the time-averaged results and the instantaneous

results. As a result, the distribution of skin friction coefficient of gap flow is nearly same with the smooth one

except for the vicinity of the gap, as shown in Fig. 9.

Gap flow of AoA =−7◦

Hypersonic aircrafts usually have a slender, streamlined fuselage to reduce the drag. In addition, the angle

of attack of the cruise phase is usually limited to several degrees. We choose a possible angle of attack to inves-

tigate how the angle of attack influences the transition process. A negative angle of attack is likely to occur on

Fig. 9 Time-averaged and spanwise-averaged distribution of skin friction coefficient at AoA = 0◦

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Fig. 10 Instantaneous slice of streamwise velocity at x-z-plane

Fig. 11 Time averaged and spanwise averaged distribution of skin friction coefficient at AoA =−7◦

the aircraft surface which means the fluid flow toward the the surface. Transition processes of AoA =−7◦ gapflow are studied in this part. Numerical results of laminar flat plate are imposed at the inflow boundary, which

are derived from a simulation of a complete geometry including the leading edge. Typical flow structures of the

transition at AoA =−7◦ are nearly the same with AoA = 0◦ case. However, the appearance of the λ vortices isearlier than AoA = 0◦ case. Contour of instantaneous streamwise velocity at the slice of y = 0.005x+ 0.0005illustrated in Fig. 10 shows a higher amplification of three dimensional instability waves after the gap.

Since the larger wave amplification in the AoA = −7◦ gap flow, a relatively earlier transition is observedcomparing to the smooth flow at AoA =−7◦.

3.4 Influence of the angle of attack on the transition process

A gap flow at AoA = 0◦ has almost the same transition process as the corresponding smooth flow. How-ever, when the angle of attack is negative, a gap can trigger an earlier transition compared to the corresponding

smooth one. In order to find out the mechanism of the interaction between forcing instability wave and gap, we

perform Fourier transformations at both angles of attack.

Mode amplitude curves in streamwise direction are plotted in Fig. 12 and Fig. 13. Modes ( f ,k) are denotedby frequency f and spanwise wave number k, where f = 1 denotes the forcing frequencies of the oblique waves(F = 0.3×10−4) and k = 1 denotes the fundamental wave number in spanwise direction (two complete waves).For each mode, Y-axis denotes the maximum streamwise velocity perturbation in boundary layer, which is

non-dimensionalized with the freestream velocity.

Modes in the gap flow at AoA = 0◦ ( Fig. 12(a) ) have similar modal developments comparing with thesmooth flow ( Fig. 12(b) ). Different modal developments are observed in the gap flow at AoA = −7◦. Theforcing modes (mode(1,1), mode(2,0)),and their second harmonic modes (mode(4,0), mode(2,2)) have the same

trend in streamwise direction as the smooth flow, which indicates the forcing modes are almost not disturbed

by the gap. However, other modes with multiple frequencies and spanwise wave numbers are excited in the gap

and develop downstream. Not only are the modes in Fig. 13(a), other frequencies and spanwise wave numbers

increase rapidly. Among all the modes, the forcing modes, mode(1,1) and mode(2,0), always dominate in the

development which determine the transition onset. This may be the reason why the change of transition onset

by a gap is not obvious.

In order to find out the reason for the gap flow behaviors of different modes, two slices of non-dimensional

Paper ID:312 8



(a) Gap, AoA = 0◦(b) Smooth, AoA = 0◦

Fig. 12 Streamwise velocity disturbance evolution at AoA = 0◦

(a) Gap, AoA =−7◦(b) Smooth, AoA = 0◦

Fig. 13 Streamwise velocity disturbance evolution at AoA =−7◦

pressure fluctuation are plotted in Fig 14. Similar to the longitudinal and transverse mechanisms proposed by

Zhang[11], two feedback mechanisms are observed in the two gap flows with different angles of attack. In the

gap flow at AoA = 0◦, the upstream disturbances propagate towards the aft face of the gap. Most disturbancesdamp at the right corner. As a result, there are no strong disturbance flow out the boundary layer. In contrast,

the disturbance path in the gap flow at AoA = −7◦ is illustrated in Fig. 14(b). There is a single feedback loopin the gap. Disturbances from the boundary layer propagate to the aft face of the gap, flow to the bottom with

the feedback loop and finally transverse back to the boundary layer. According to the experimental results[10],

a longitudinal mechanism exists in a shallow gap and a deep gap leads to a transverse mechanism. The results

in this paper show that a negative angle of attack can also cause a transition from longitudinal to transverse

mode besides the length-to-depth ratio. This may be the reason for the occurrence of the broadband distur-

bance observed in the FFT analysis. Unfortunately, the typical frequencies for the two mechanisms can not

be extracted with the current frequency analysis. A large number of instability waves are produced with this

nonlinear interaction, causing the sudden rise in mode amplitudes, as shown in Fig.13(a).

(a) Pressure fluctuation in gap flow at AoA = 0◦ (b) Pressure fluctuation in gap flow at AoA =−7◦

Fig. 14 Distribution of pressure fluctuation at the slice of z = 0.009m

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4 Conclusions

Transition processes of a Mach 6 boundary layer with a 2D gap have been studied for two angles of attack,

AoA = 0◦ and AoA = −7◦ with a high order ILES tool, hpMusic, in the present paper. Both 2D and 3Dinstability waves were introduced at the inflow to obtain a subharmonic transition in the simulations. Some

conclusions are drawn next.

In the case of AoA = 0◦, the transition phenomenon of the gap flow is similar to that on smooth one. Theinfluence of the gap occurs mainly in the vicinity of the gap. The flow outside the gap is almost undisturbed.

There is no extra disturbance observed in the gap, and the evolution of the disturbances downstream the gap

is similar to that of the smooth flow. Instability modes propagate toward the aft surface of the gap and damp

rapidly. As a result, little difference is observed in the distribution of skin friction coefficient between gap flow

and smooth flow.

The forcing disturbances show a different interaction with the gap at AoA =−7◦. Transition onset is earlierthan the corresponding smooth one. Three dimensional vortex structure in the gap flow occurs earlier than that

in the smooth flow. According to the FFT analysis, plenty of instability modes other than the forcing modes or

the modes originated from the forcing modes, are observed in the gap and flow downstream which contribute

to transition. A single feedback loop is observed within the gap which is considered to be the reason for the

jump of the broadband modes at the gap.

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Paper ID:312 11

IntroductionNumerical MethodSimulation results and discussionsConclusions

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Impact of a gap on boundary layer transition at Mach 615th International Conference on Fluid Control, Measurements and Visualization 27-30 May, 2019, Naples, Italy Impact of a gap