# Suppression of the Spatial Hydrodynamic Instability in Scale-Resolving Simulations of Turbulent Flows Inside Lined Ducts

^{*}

## Abstract

**:**

## 1. Introduction

**n**is assumed to point inside the liner, R and X are the acoustic resistance and reactance, respectively, $i\equiv {(-1)}^{1/2}$, and the time-dependence of the harmonic signal is assumed to have the form $\mathrm{exp}(-i\omega t)$.

_{e}) is equal to 1000 Hz, which value is close to the resonant frequency f

_{RES}= 1040 Hz of the tested ceramic tubular liner CT57 (at this frequency, the liner’s normalized acoustic resistance, $R/(\rho c)$, reaches the minimum of ~0.4), and manifests itself in the formation of periodic large-scale structures in the close vicinity of the lined wall. These structures have a rather large correlation length in the spanwise direction (they are nearly 2D) and are convecting along the wall at a velocity of about 70% of the bulk flow velocity, with the amplitude of the instability wave growing while moving downstream. This causes a substantial alteration of both the turbulence characteristics (intensity of the fluctuations and their spectra) and the parameters of the mean flow in the near-wall area. Judging by the experimental distributions of the sound pressure level (SPL) along the duct wall [14], in the real GIT flow, a minor instability must be present as well. However, the WMLES [13] drastically overemphasizes its growth rate, which results in quite poor agreement with the experiment. Moreover, in the simulations, the instability waves at the resonant frequency are observed not only in the case of the near-resonant external forcing but also at all the other tested forcing frequencies and even in the case when there is no forcing signal. These prominent features of the hydrodynamic instability found in [13] are illustrated by sample results from the simulations [13] in Figure 1, Figure 2 and Figure 3 included in the present paper for the sake of self-containment. The instability originates from a very narrow area adjacent to the lined wall and supposedly has an inviscid nature resembling the Kelvin-Helmholtz instability [16,17]. It should be emphasized that the instability observed in the WMLES solutions [13] has a physical rather than numerical origin, and its characteristics are not affected significantly by a strong grid-refinement.

## 2. Flows under Consideration and Some Numerical Details

#### 2.1. Ducts Geometries and Flow Regimes

_{y}= L

_{z}= H = 0.0508 m and a length of L

_{x}= 0.8228 m. The liner sample covers a part of the upper wall of the test section, starting at x = 0.02032 m and ending at x = 0.6096 m. The plane sound waves in the experiment are introduced at the inlet of the test section (at x = 0), and their intensity is automatically adjusted to ensure that the SPL at the center of the lower wall at x = 0 is equal to 130 dB. The acoustic pressure is measured along the centerline of the lower wall of the test section. The measurements are performed at six different frequencies of the external acoustic forcing f

_{e}within the range (500–3000) Hz with a 500 Hz increment.

_{AV}= 0.335 in the reference measurement cross-section (x = 0.356 m), for which the measured Mach number field is available in [14]. It should be noted that the published NASA GIT experimental papers [14,34] do not provide the duct geometry upstream of the test section, which makes accurate reproduction of the experimental velocity profiles in the simulations problematic. An investigation and discussion of the sensitivity of the predictions to the shape of the inflow velocity profile can be found in [13].

_{x}= 1.27 m with the cross-section dimensions L

_{y}× L

_{z}= 0.0635 m × 0.0508 m. The liner sample is placed on the upper wall at 0.284 m ≤ x ≤ 0.69 m. Similar to the GIT experiment, the SPL of the waves propagating inside the GFIT test section is measured along the centerline of the lower duct wall opposite the liner. In addition, some measurements are performed on its upper wall outside of the lined part of the surface. The measurements in [32] are carried out for the external acoustic forcing (injection of the plane sound waves) at a number of frequencies within the range (400–3000) Hz. The intensity of the injected waves at x = 0 (at the location of the first microphone) is equal to 140 dB.

_{AV}= 0.362, and within the test section, it gradually increases up to the value of ~0.4 at x, corresponding to the middle of the liner.

#### 2.2. Computational Problem Statement

_{AV}= 0.335 at x = 0.356 m for the GIT [14] and M

_{AV}= 0.362 at x = −0.143 m for the GFIT [32], see Figure 4).

_{e}and intensities (SPL), whose propagation through the ducts is computed in the production stage of the simulations, are “created” at the URANS inflow of the computational domain by means of the characteristic boundary conditions formulated via the 1-D Riemann invariants. Specifically, harmonic pressure fluctuations with the prescribed SPL value and corresponding fluctuations of the streamwise velocity and density are imposed on top of the mean inlet flow quantities, and thus obtained unsteady quantities are used to set the time-dependence of the incoming Riemann invariants ${I}_{1}=u+2c/(\gamma -1)$ and ${I}_{5}=p/{\rho}^{\gamma}$ (u is the streamwise velocity, c is the local speed of sound, p and $\rho $ are the static pressure and the density, respectively, and $\gamma $ is the specific heats ratio assumed to be constant equal to 1.4). The outgoing invariant at the inflow boundary ${I}_{2}=u-2c/(\gamma -1)$ is computed using extrapolation from the interior of the domain, and the transverse velocity components (the invariants ${I}_{3}$ and ${I}_{4}$) are set zero (v = w = 0).

_{e}≤ 3000 Hz) demonstrated in [13,31], and by a direct comparison (not shown) of the predictions of the SPL(x) distributions along the GIT wall obtained in the present work in the framework of the zonal URANS-IDDES combined with the quasi-2D and fully 3D problem statements.

#### 2.3. Impedance Approximations, Numerics, and Grids

_{AV}= 0.4, i.e., at the value of the section-averaged Mach number in the middle of the test section for the GFIT flow regime computed in the present work (see Section 2.1 above). In addition, as already mentioned, the SDoF liners are subject to a non-linear dependence of their acoustic impedance on the sound wave intensity. Although for the AE02 liner at M

_{AV}= 0.4, this effect is relatively weak, it is not negligible and is accounted for in the simulations thanks to the dynamic impedance capability of the TDIM [13]. Specifically, the simulations are carried out with the use of three reference levels for the root-mean-square wall-pressure fluctuations, ${p}_{RMS}^{(l)}$ = 130, 140, and 150 dB. Corresponding approximations of the liner resistance and reactance as functions of frequency obtained with the use of the VF method are presented in Figure 7. At each ${p}_{RMS}^{(l)}$ value, these approximations include 6 poles, resulting in 18 poles total in the expression (6) for the “aggregated” dynamic impedance. Therefore, at all the wall grid-nodes located within the entire lined surface, it is necessary to solve 18 auxiliary ODEs (5).

^{−3}m, i.e., somewhat smaller than 1/10-th of the ducts’ half-height H/2, which can be interpreted as the thickness of the boundary layer ${\delta}_{BL}$ for the developed channel flow. Downstream of the area of interest (within the outlet sponge layer), in order to reduce the total number of cells and to ensure a more efficient suppression of turbulent and acoustic disturbances, $\mathsf{\Delta}x$ is gradually increased up to 0.01 m. The maximum grid step in the wall-normal direction, ${(\mathsf{\Delta}y)}_{\mathrm{max}}$, is set equal to $\mathsf{\Delta}x/2$ = 1.25 × 10

^{−3}m (less than ${\delta}_{BL}/20$). In addition, the grids in the y-direction are strongly clustered towards the walls with the near-wall step $\mathsf{\Delta}{y}_{1}=5\times {10}^{-6}$ m, which ensures its value is less than one in the wall units. The grids in the spanwise direction are uniform, with the step $\mathsf{\Delta}z={(\mathsf{\Delta}y)}_{\mathrm{max}}$. This results in a grid of ${N}_{x}\times {N}_{y}\times {N}_{z}$ = $620\times 102\times 44$ cells for the GIT [14] flow (~2.8 million cells total) and ${N}_{x}\times {N}_{y}\times {N}_{z}$ = $751\times 112\times 44$ (~3.7 million total) cells for the GFIT [32] flow. Note that in terms of the spatial resolution of the sound waves within the frequency range of interest (f ≤ 3000 Hz), these grids, which are built based on the WMLES demands, guarantee also very accurate representation of the sound propagation because they provide more than 40 cells per wave-length, whereas the high-order numerical scheme used in the simulations ensures an acceptable accuracy even with 10 cells per wave-length.

## 3. Design of Novel Stabilization Body Force Based on Simulations of NASA GIT Flow over a Ceramic Tubular Liner

#### 3.1. General Consideration and Shortcomings of PGTS Method

_{D}in (7) is related to the parameter ε of [17] by the formula B

_{D}= 1 − ε (the value of ε = 1 and so B

_{D}= 0 corresponds to the case of the original governing equations). The remaining quantities are defined as follows: x and y are the streamwise and wall-normal coordinates, u, and v are the instantaneous values of corresponding velocity components, ρ is the instantaneous density, and $\overline{\rho}$, $U$, and V are the mean values of the corresponding functions. Note that if the damping source term SRC1 is used for stabilization of the simulations in the framework of the non-linear flow models rather than in the framework of the LEE (as in [17]), the mean flow parameters entering (7) are a priori unavailable and thus should be computed in the course of the simulation. For this purpose, we have used a sliding average of the unsteady solution.

_{AV}= 0.335 grazing flow [14] with no external acoustic signal and compared the obtained solution with that of the similar computations [13] with the use of the original governing equations for both the solid-wall and the lined-wall ducts. Recall that, in spite of the absence of external forcing, the WMLES of the flow with the liner predicts the development of a strong hydrodynamic instability resulting in a strong alteration of even the mean flow characteristics (see Figure 1 and Figure 2 in Section 1).

_{D}in (7) was identified via a series of computations with different B

_{D}-values. Its minimum for ensuring complete suppression of the instability turned out to be equal to 0.40, which is virtually the same as the stability boundary for this flow predicted by the quasi-1D LSA [24] based on the linearized Navier-Stokes equations.

_{D}are presented in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 in the form of the instantaneous fields (Figure 8 and Figure 9), the spectra of the pressure at the wall at x = 0.6 m, i.e., at the location close to the downstream edge of the liner (Figure 10), the streamwise distributions of the span-averaged root-mean-square fluctuations of the wall pressure and the time-averaged friction coefficient (Figure 11), and, finally, the wall-normal distributions of the root-mean-square pulsations of the velocity components and the resolved turbulent kinetic energy (TKE) at x = 0.6 m (Figure 12). The figures confirm the stabilizing capability of the PGTS method, but at the same time, they disclose a far too strong damping of the turbulent pulsations by the source term SRC1 (7), especially in the high-frequency range. This results in a substantial deterioration of the spectra of the wall-pressure and of the skin-friction distribution. Moreover, the maximum of the TKE in the near-wall region is reduced by more than 30% compared to the solid-wall duct. This suggests that the PGTS source term, proven in [17] to be fairly efficient in terms of suppressing the instability in the LEE computations, performs unsatisfactorily in the framework of the scale-resolving approaches.

#### 3.2. Design of New Body Force

_{s}being a non-dimensional constant (the amplitude factor) depending on the impedance of the liner under consideration.

_{s}was set equal to 0.02, which value is close to the stability boundary B

_{s}= 0.019 obtained from the LSA [24]). Indeed, with the new body force, the maximum and the entire profiles of the velocity fluctuations and the TKE differ from the corresponding quantities in the target solid-wall duct simulation much less than those with the body force SRC1. Additionally, as seen in Figure 14, the same conclusion can be drawn regarding the deviations of the wall-pressure spectrum, the root-mean-square of the pressure pulsations, and the mean friction coefficient.

_{D}= 0.4 for the PGTS body force (7) and B

_{s}= 0.02 for the proposed body force (9), that is, at their minimum values, allowing elimination of the instability developing in the considered flows. The Table shows two complex eigenvalues k

_{x}(normalized with the half-width of the duct) corresponding to the sound waves of different frequencies propagating upstream and downstream (k

_{1}and k

_{2}, respectively) predicted by the LSA [24] for the unchanged (with no artificial source terms) Navier–Stokes equations and for these equations with the added source terms SRC1 and SRC2. The real part of the eigenvalues is the axial wavenumber, and the imaginary part (with the sign depending on the propagation direction) is the growth rate of the disturbances. The data in the table suggest that the body force SRC2 leads to a weaker alteration of the acoustic modes and, therefore, should have a weaker effect on the prediction of sound propagation. This LSA-based conclusion is supported by the results of numerical experiments presented in Section 3.3 below. Thus, the proposed body force has an advantage over the PGTS one not only in terms of turbulence treatment but in terms of acoustics as well.

_{s}is the width of the non-zero body force region, which is assumed to be small compared to the boundary layer thickness at the lined wall.

_{s}was set equal to 0.01 of the GIT half-height H/2 (or the ${\delta}_{BL}$), and its amplitude B

_{s}was set equal to 0.04, i.e., was increased by a factor of 2 compared to the simulation with the non-clipped body force (9). The figure clearly suggests that when the stabilizing body force is deactivated at ${d}_{w}>0.01{\delta}_{BL}$, the predicted maximum of the TKE in the boundary layer virtually coincides with that in the duct with the solid walls, and that the area of a noticeable effect of the body force on the TKE field is confined by the near-wall zone of the width about $0.05{\delta}_{BL}$. As for the mean friction coefficient (not shown), it differs from that for the solid-wall case by only ~3% versus ~8% for the non-clipped body force SRC2 (9).

#### 3.3. Calibration of the Body Force Parameters and Its Effect on Propagation of Sound Waves at Non-Resonant Frequencies

_{s}and the amplitude factor B

_{s}, we have used the NASA GIT experiment with CT57 liner [14] in the presence of external forcing at the near-resonant frequency f

_{e}= 1000 Hz. For this case, the WMLES [13] predicts the existence of hydrodynamic instability with a spatial growth rate much higher than that in the experiment (see [13] and Figure 1, Figure 2 and Figure 3 in Section 1). Therefore, this is the most suitable case for the calibration of the constants in the stabilization source term.

_{s}is concerned, the inviscid origin of the hydrodynamic instability developing along the lined wall suggests that it should be scaled with the “vorticity thickness” of the near-wall velocity profile defined as ${\delta}_{\omega}={U}_{e}/|\partial U/\partial y{|}_{w}$, where ${U}_{e}$ is the boundary layer edge velocity (for the developed flow in a duct, it is replaced by the bulk velocity). In the course of the calibration, we varied the ratio ${W}_{s}/{\delta}_{\omega}$ in the range $1.5<{W}_{s}/{\delta}_{\omega}<6$. For the GIT flow, this corresponds to the variation of ${W}_{s}$ from 0.005 up to 0.02 of the half-height of the duct. Thus, the region of non-zero source terms in all the considered cases was very narrow.

_{s}was varied from 0.02 up to 0.1, which values were deliberately chosen to be too small and too large, respectively.

_{s}needed for the elimination of the instability in the case under consideration. As a result, the main part of the wall-normal profile of the TKE, including its near-wall maximum, is virtually unaffected by the damping source term.

_{e}= 500 Hz, 1500 Hz, and 3000 Hz, covering the entire range of the frequency variation in the experiments (500 Hz < f

_{e}< 3000 Hz), with the stabilization body force (10) “turned on”. Its parameters were set as ${W}_{s}=3{\delta}_{\omega}$ and ${B}_{s}=0.06$, i.e., equal to the values just proven to enable elimination of the strong instability observed at the near-resonant forcing frequency. In Figure 22, corresponding SPL distributions along the lower wall of the GIT test section are compared with the similar distributions [13] obtained in the WMLES of the same cases carried out using the original (with no artificial source terms) governing equations and with the experiment [14].

_{e}= 3000 Hz (Figure 22c) downstream of x = 0.5 m, where the sound wave intensity becomes so low that the computed SPLs are almost entirely defined by the “turbulent floor” of the wall-pressure pulsations.

_{s}and the vorticity thickness at the lined wall δ

_{ω}chosen for the proposed stabilizing body force (10) based on the WMLES of the flow in the GIT with the CT57 liner $({W}_{s}/{\delta}_{\omega}=3)$ can be considered quite universal, i.e., can be used for suppressing the spatial instability in the scale-resolving simulations of flows over any other liners characterized by low acoustic resistance, when such instability may show up. As for the second parameter of the body force (the amplitude factor B

_{s}), its minimum value ensuring elimination of the instability depends on the liner impedance and, therefore, should be adjusted to every specific liner. An example supporting these statements is presented in the next section.

## 4. Application to an Alternative (SDoF) Liner

_{e}= 1400 Hz. Judging by the URANS computations [31], in this flow, the hydrodynamic instability at the lined surface develops not only at the natural (resonant) frequency of the AE02 liner (f

_{RES}≈ 1600 Hz), but simultaneously at the forcing frequency itself. For this reason, in terms of suppressing the instability in the simulations, this case is the most challenging one.

## 5. Conclusions

_{s}), whose optimal value depends upon the liner impedance at the resonant sound frequency. This task is facilitated by the weak sensitivity of predictions to an increase in B

_{s}versus its minimum value, ensuring a complete suppression of the hydrodynamic instability.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**WMLES predictions [13] of the effect of liner and acoustic forcing on instantaneous fields of vorticity magnitude in a meridian plane and at the upper (lined) wall and pressure fluctuations in a meridian plane of NASA GIT [14] (liner shown by grey rectangular is located at 0.2 m < x < 0.61 m). (

**a**): flow in solid-wall duct (no liner); (

**b**): flow in duct with liner and acoustic forcing at f

_{e}= 1000 Hz; (

**c**): flow in duct with liner without any external forcing.

**Figure 2.**Effect of liner and acoustic forcing on the wall-pressure spectrum at a location close to liner’s downstream edge (

**a**) and mean skin-friction distribution along duct upper wall (

**b**) predicted by WMLES [13]. Green line: solid-wall duct (no liner); red: duct with liner and external forcing by plane sound waves of intensity 130 dB at f

_{e}= 1000 Hz; blue: duct with liner and without any external forcing.

**Figure 5.**Schematic of XY-cut of the computational domain used in zonal RANS-IDDES showing locations of the VSTG and the exit sponge layer together with an example of an instantaneous field of vorticity magnitude.

**Figure 6.**Frequency-dependence of acoustic resistance R and reactance X of the CT57 liner from experiment [14] and its VF-approximation by multipole function.

**Figure 7.**Acoustic resistance and reactance of AE02 liner at M

_{AV}= 0.4 and different levels of wall-pressure fluctuations.

**Figure 8.**Instantaneous fields of vorticity magnitude at the upper (lined) wall and pressure fluctuations in a meridian plane from WMLES of the flow in the GIT without external acoustic signal with the use of stabilization body force SRC1 (7) at B

_{D}= 0.4. Compare these fields with the corresponding fields in Figure 1a,c.

**Figure 9.**Instantaneous fields of fluctuations of vertical (normal to the lined wall) velocity at x = 0.6 m from WMLES of flows in the duct with solid walls using original governing equations (

**a**) and in the duct with liner using equations with added stabilization body force SRC1 (7) at B

_{D}= 0.4 (

**b**).

**Figure 10.**Comparison of wall-pressure spectra at x = 0.6 m predicted by WMLES of three flows. Green: duct with solid walls, original equations; red: duct with liner, original equations; blue: duct with liner, equations with added stabilization body force SRC1 (7) at B

_{D}= 0.4.

**Figure 11.**Same as in Figure 10, for streamwise distributions of root-mean-square wall-pressure fluctuations (

**a**) and mean friction coefficient (

**b**).

**Figure 12.**Comparison of wall-normal profiles of root-mean-square pulsations of velocity components and resolved TKE at x = 0.6 m predicted by the three WMLES.

**Figure 13.**Comparison of wall-normal profiles of root-mean-square pulsations of velocity components and resolved TKE at x = 0.6 m from simulations of flows in the solid-wall duct using original governing equations; in the duct with liner using equations with added stabilization body force SRC1 (7) at B

_{D}= 0.4; and in the duct with liner using equations with added stabilization body force SRC2 (9) at B

_{s}= 0.02.

**Figure 14.**Same as in Figure 13, for wall-pressure spectra at x = 0.6 m (

**a**) and streamwise distributions of root-mean-square wall-pressure pulsations (

**b**) and mean friction coefficient (

**c**).

**Figure 15.**Comparison of wall-normal profiles of resolved TKE at x = 0.6 m predicted for solid-wall duct using original governing equations and for lined duct using equations with added clipped stabilizing body force SRC2 (10).

**Figure 16.**Instantaneous field of pressure fluctuations in a vertical plane XY from simulation of NASA GIT flow at f

_{e}= 1000 Hz using optimal (${W}_{s}=3{\delta}_{\omega}$, ${B}_{s}=0.06$ ) stabilization source term (10). Compare this field with similar field from WMLES without stabilization in Figure 1b.

**Figure 17.**Effect of stabilization (10) with optimal parameters on downstream evolution of pressure spectra at the lined wall for NASA GIT flow at f

_{e}= 1000 Hz. Red line: computation without stabilization [13]; black: with stabilization; green: flow in solid-wall duct.

**Figure 18.**Effect of stabilization (10) with optimal parameters on SPL-distribution along lower (

**a**) and upper (

**b**) walls of GIT.

**Figure 19.**Effect of stabilization (10) with optimal parameters on wall-normal profiles of root-mean-square pulsations of velocity and resolved TKE at x = 0.6 m.

**Figure 20.**Effect of stabilization (10) with optimal parameters on skin-friction at lined wall of GIT.

**Figure 21.**Sensitivity of predictions of NASA GIT flow at f

_{e}= 1000 Hz to parameters B

_{s}and W

_{s}of stabilization body force (10). Upper row: spectra of wall-pressure fluctuations at x = 0.6 m; mid row: SPL at the lower wall; lower row: profiles of resolved TKE at x = 0.6 m.

**Figure 22.**Effect of stabilization (10) on computed SPL distributions along the GIT wall at non-resonant forcing frequencies. (

**a**): f

_{e}= 500 Hz; (

**b**): 1500 Hz; (

**c**): 3000 Hz.

**Figure 23.**Instantaneous iso-surface of swirl parameter (magnitude of the second eigenvalue of the velocity gradient tensor) ${\lambda}_{2}=10{c}_{0}/(H/2)$ from the simulation of the flow in NASA GFIT with AE02 liner using the original governing equations. The iso-surface is “colored” by local value of streamwise velocity component.

**Figure 24.**Effect of stabilization (10) and its amplitude parameter B

_{s}on instantaneous field of pressure fluctuations in a vertical plane of NASA GFIT flow. Location of AE02 SDoF liner on the upper wall is shown by the grey rectangular.

**Figure 25.**Effect of stabilization (10) and its amplitude parameter B

_{s}on the evolution of wall pressure spectra at the upper (lined) wall of NASA GFIT.

**Figure 26.**Effect of stabilization (10) and its amplitude parameter B

_{s}on distributions of SPL along lower and upper walls of NASA GFIT.

**Figure 27.**Effect of stabilization (10) and its amplitude parameter B

_{s}on wall-normal profiles of root-mean-square pulsations of velocity and resolved TKE at section x = 0.6 m of NASA GFIT flow.

**Figure 28.**Effect of stabilization (10) and its amplitude parameter B

_{s}on mean skin-friction coefficient at upper wall of NASA GFIT.

**Table 1.**LSA predictions of acoustic complex eigenvalues k

_{x}for original Navier-Stokes equations and equations with added stabilization source terms SRC1 (7) and SRC2 (9).

f, Hz | k_{1} | k_{2} | ||||
---|---|---|---|---|---|---|

Unchanged Equations | with SRC1 | with SRC2 | Unchanged Equations | with SRC1 | with SRC2 | |

500 | 0.235 + 0.020i | 0.242 + 0.021i | 0.238 + 0.023i | −0.642 − 0.114i | −0.593 − 0.096i | −0.621 − 0.100i |

1000 | 0.432 + 0.281i | 0.431 + 0.323i | 0.427 + 0.295i | −0.406 − 0.486i | −0.432 − 0.478i | −0.415 − 0.489i |

1500 | 0.449 + 0.058i | 0.443 + 0.062i | 0.447 + 0.058i | −0.863 − 0.110i | −0.877 − 0.103i | −0.868 − 0.111i |

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## Share and Cite

**MDPI and ACS Style**

Shur, M.; Strelets, M.; Travin, A.
Suppression of the Spatial Hydrodynamic Instability in Scale-Resolving Simulations of Turbulent Flows Inside Lined Ducts. *Fluids* **2023**, *8*, 134.
https://doi.org/10.3390/fluids8040134

**AMA Style**

Shur M, Strelets M, Travin A.
Suppression of the Spatial Hydrodynamic Instability in Scale-Resolving Simulations of Turbulent Flows Inside Lined Ducts. *Fluids*. 2023; 8(4):134.
https://doi.org/10.3390/fluids8040134

**Chicago/Turabian Style**

Shur, Mikhail, Mikhail Strelets, and Andrey Travin.
2023. "Suppression of the Spatial Hydrodynamic Instability in Scale-Resolving Simulations of Turbulent Flows Inside Lined Ducts" *Fluids* 8, no. 4: 134.
https://doi.org/10.3390/fluids8040134