Shock System Dynamics of a Morphing Bump Over a Flat Plate

TL;DR

This study investigates shock dynamics caused by a moving bump over a flat plate at Mach 2.9, using numerical simulations to identify optimal bump speeds and transition times for minimal entropy change in supersonic flow.

Contribution

It introduces a numerical approach to analyze shock system dynamics of a morphing bump in supersonic flow and determines optimal movement parameters for minimal flow disturbance.

Findings

Spatial lag in shock system increases with bump speed.

Optimal non-dimensional time for minimal entropy change is 2.

Moving the bump for too long causes flow deviations from steady state.

Abstract

In this paper, the shock dynamics due to the movement of a bump over a flat plate flying at supersonic speed are numerically investigated. The bump is located at the impingement position of the shock wave and is moved at different speeds. This study determines the suitable speed that achieves the minimum entropy change, which is the representation parameter of the transition period. The two-dimensional unsteady Navier-Stokes equations are solved using OpenFOAM to simulate the flow field variables, while the motion of the bump is tracked using the Arbitrary Lagrangian-Eulerian (ALE) technique. The results show that a spatial lag on the shock system from the steady-state solution occurs due to the movement of the bump. Further, the spatial lag increases with the increase in the bump's speed. This causes a high increase in the flow parameters and consequently the total entropy changes on the bump surface. Generally, it is common to move the bump over the longest possible time to approximate a quasi-steady flow during the motion. However, this causes a deviation in the flow parameters between the final time of transition and the steady-state case of bump existence. Thus, it is concluded that the optimal non-dimensional time for a morphing bump in a supersonic flow of Mach number of 2.9 is 2, which is different than the longest time of 10.

Tables

Table 1. Table 1 : System properties and parameters.

Mach number, $M_{\infty }$	$2.9$
Reynolds number, $R{e}_{\infty }$	$6.6\times {10}^7$
Inlet length, $l_i$	$1.1m$
Wedge chord, $c_w$	$0.527m$
Wedge angle, ${\alpha }_w$	${11}^{\circ }$
Slip wall length, $l_{sl}$	$0.26555m$
Bump final length, $l_b$	$0.5m$
Bump final height, $h_b$	$0.01m$
Motion periods, $t_f^{\ast }$	$0.1$, $1$, $1.8$, $2$, $3$, $5$, $7.5$, $10$

Equations

$$y=\sqrt{{\left(\frac{{\alpha }^2+0.25}{2\alpha }\right)}^2-x^2}+\frac{{\alpha }^2-0.25}{2\alpha }$$

$$\frac{\partial \rho }{\partial t}$$

$$\rho \left(\frac{\partial \text{V}}{\partial t}+\left(\text{V}\cdot \nabla \right)\text{V}\right)$$

$$=-\nabla p+\nabla \cdot$$

$$\rho \left(\frac{\partial e}{\partial t}+\left(\text{V}\cdot \nabla \right)e\right)$$

$$=-\frac{2}{3}\mu$$

Full text

**Conference Paper

2022

**

**Shock System Dynamics of a Morphing Bump Over a Flat Plate

**

**Ahmed A. Hamada1,2, Lubna Margha1,2, Mohamed M. AbdelRahman2, Amr Guaily3,2

1Department of Ocean Engineering, Texas A$\&$M University, College Station, TX, 77843, USA

2Department of Aeronautical and Aerospace Engineering, Cairo University, Giza, 12613, Egypt

3Smart Engineering Systems Research Center (SESC), Nile University, Giza, 12588, Egypt

**

Abstract

The shock wave boundary layer interaction (SW-BLI) phenomenon over transonic and supersonic airfoils captured the attention of aerospace engineers, due to its disastrous effect on the aerodynamic performance of these vehicles. Thus, the scientific community numerically and experimentally investigated several active and passive flow control elements to reduce the effect of the phenomenon, such as vortex generator, cavity, and bump. They focused on designing and optimizing the shape and location of the bump control element. However, the transit movement of the bump from the state of a clean airfoil to the state of an airfoil with a bump needs more investigation, especially the dynamics of the shock system. Thus, it is preferred to start with simple geometry, such as a flat plate, to fully understand the flow behavior with a morphing bump. In this paper, the shock dynamics due to the movement of a bump over a flat plate flying at supersonic speed are numerically investigated. The bump is located at the impingement position of the shock wave and is moved at different speeds. This study determines the suitable speed that achieves the minimum entropy change, which is the representation parameter of the transition period. The two-dimensional unsteady Navier-Stokes equations are solved using OpenFOAM to simulate the flow field variables, while the motion of the bump is tracked using the Arbitrary Lagrangian-Eulerian (ALE) technique. The results show that a spatial lag on the shock system from the steady-state solution occurs due to the movement of the bump. Further, the spatial lag increases with the increase in the bump’s speed. This causes a high increase in the flow parameters and consequently the total entropy changes on the bump surface. Generally, it is common to move the bump over the longest possible time to approximate a quasi-steady flow during the motion. However, this causes a deviation in the flow parameters between the final time of transition and the steady-state case of bump existence. Thus, it is concluded that the optimal non-dimensional time for a morphing bump in a supersonic flow of Mach number of 2.9 is 2, which is different than the longest time of 10.

Keywords: Moving bump; Supersonic flow; Flat plate; Active flow control.

1.   Introduction

The Shock wave boundary layer interaction (SWBLI) over a high-speed wing disastrously affects its aerodynamics performance and structural lifetime. The phenomenon of SWBLI was experimentally observed for the first time by Ferri [1] in 1939. Green [2] summarized the occurrence of the phenomenon in four conditions: externally as in transonic airfoils, and near control flaps, or internally as in high-speed inlets (scram-jets), and nozzles at off-design. Based on strength of the shock wave, the

phenomenon varies the height of the boundary layer, the surface drag of the body, and/or the heat transfer. The SWBLI adversely affects the structure and geometry of the flying vehicles [3]. The flow experiences a high instantaneous increase in pressure and thermal transfer due to the existence of SWBLI, which leads to a reduction in the fatigue life of the structure. Consequently, that imposes strong constrictions on choosing the material of the structure, resulting in an expensive and heavy design. Moreover, this phenomenon happens in transonic airfoils after a critical Mach number, causing the sudden increase in total drag, which is called “drag divergence”.

There are several ways to eliminate or reduce the SWBLI effect which is discussed in detail by Dennis Bushnell [4]. The focus in this paper will be on the bump approach. In 1922, Ashill et al. [5] were the first researchers who used a two-dimensional bump at the upper surface of airfoils to affect the strength of incurring shock waves. Milholen et al. [6] and Patzold et al. [7] proved that using bumped airfoils increases the lift, reduces the drag, and postpones the buffeting, which enhances the aerodynamic performance. Fulker [8], who is a DASA-Airbus researcher, showed that applying a bump in A-340’s hybrid laminar wing will save fuel by $2.11\%$ at $M_{\infty}=0.84$. Several investigations are conducted to conclude with the optimal shape and location of the bump [9, 10, 11, 12]. Further, Eastwood and Jarrett [13] investigated the three-dimensional bump control technique. Yun Tian et al. [14] showed that airfoil geometry and free stream condition determine the bump’s optimal geometric parameters which specify its shape and location over the airfoil. In addition, varying these geometric parameters highly affects the aerodynamic performance.

By the beginning of the 21st century, the morphing wing concept become industrial applicable. Stanewsky et al. discussed the design aspects of the morphing concept from the view of the civil aircraft industry [15]. From that time, many scholars started to investigate the flow behavior over the morphing wing at different velocity regimes [16, 17, 18, 19]. Further, Bruce and Colliss [20] conducted a review study for the shock control bumps, showing that the morphing bump is a future ideal solution to limit the SWBLI effect on wings at transonic flows.

To simplify the phenomenon, consider a strong two-dimensional shock wave that impinges with a boundary layer of a flat plate, see Figure 1. Downstream the impinging region, separation, and relaxation zones are generated. The flow changes from a non-equilibrium state in the separation zone to an equilibrium state in the relaxation zone. Through these zones, the flow experiences a quick increase in pressure and heat transfer, detaching, and reattaching the boundary layer. Moreover, the turbulent kinetic energy increases due to the generation of the strong shear and the adverse pressure gradient in and near the separation zone, respectively. During the phenomenon, the flow contains different types of waves generated from the SW-BL.I, such as; a reflected shock wave, expansion waves, and compression waves [21].

The static condition for the bump control element is fully investigated. Particularly, they concentrated on designing and optimizing the bump control element’s shape and position. However, additional detailed investigations are needed for the transition of the morphing bump from a clean airfoil to an airfoil with a bump, focusing on the dynamics of the shock system. The aim of this research is to numerically study the transient effect of the morphing pump over a simple geometry (flat plate) on the unsteady flow features of the shock dynamics. We started to study the transient phenomenon over a simple geometry first to fully understand the transition effects. The location of the bump is chosen to be at the impingement location of the incident shock wave. Then, starting the unsteady motion of the bump at different speeds and a constant velocity profile. This enabled us to determine the suitable morphing bump’s speed that would minimize the entropy change.

2.   Physical Model

2.1 Model Description

The flow configuration of the morphing bump in a supersonic flow is shown in Figure 2. Further, the variation of the morphing bump shape with time is shown in Figure 3. The height of the computational inflow boundary is $1.1m$, and the height of the outflow at the right is $1m$. The difference in height between the two edges represents the height of the wedge, which is the shock-source surface. The wedge is far from the inlet with a length of $0.2656m$ and the wedge stream-wise length is $0.5173m$. The morphing bump is placed at the center of the bottom surface, representing the origin of the domain. The origin is $2.25m$ far away from the inlet. The final shape of the morphing bump is a parabolic arc of height-to-length ratio, $\alpha=4\%$, and represented by Equation (1).

[TABLE]

When the supersonic flow with a free-stream Mach number $M_{\infty}=2.9$ and a free-stream Reynolds number $Re_{\infty}=6.6\times 10^{7}$ impinges the wedge, an incident shock wave (I) is generated hitting the bottom surface. Then, a reflected shock wave (R) appears. Further, a series of expansion fans generates from the wedge trailing edge (E). The expansion fan interacts with the shock system causing a curvature to them. The remaining top surface after the wedge is considered an outlet to enable the flow to go out freely. The morphing bump rises to its final shape with a constant velocity profile and different speeds (different motion periods of $0.1$, $1.0$, $5.0$, $7.5$, $10.0$). The initial condition of the problem is shown in Figure 2 (a) with the solid lines, while the dashed lines are for the final condition. The values of the problem’s parameters are shown in Table 1. The appearance of the morphing bump causes a generation of two oblique shocks at its leading edge (L) and at its trailing edge (T). Further, a series of expansion fans appear over the morphing bump due to its geometric curvature (Series of Es).

2.2 Governing Equations

Two-dimensional unsteady compressible Navier-Stokes equations are used to model the supersonic flow, and are expressed as:

[TABLE]

where $\rho$ is the density, $V$ is the velocity vector, $p$ is the pressure, $e$ is the specific energy per unit mass, $\mu$ is the fluid dynamic viscosity.

3.   Computational Model

3.1 Numerical Implementation

rhoCentralDyMFoam is a toolbox flow solver in OpenFOAM®$-$v2006. It solves transient, compressible flow problems using a density-based approach, supporting dynamic mesh applications. The semi-discrete and upwind-central non-staggered schemes of Kurganov and Tadmor are combined in the solver [22, 23], which uses an operator-splitting method [24]. The van Leer limiter is implemented in the solver to balance between shock capture, oscillations-free fields, and computational cost during the solution [25].

The letters ”DyM” in the name of the rhoCentralDyMFoam solver indicates that the dynamic mesh simulation is applicable with a dynamic FV mesh technique. In this study, the motion of the bump will be actively controlled. Thus, the coded motion solver is used in this study to arbitrarily move the computational mesh inside the domains, optimizing the cells’ shape. Hence, the dynamic FV mesh follows the Arbitrary Lagrangian-Eulerian technique, which is presented by Hirt et al. [26].

3.2 Computational Domain

A body-fitted computational domain is constructed without the existence of the bump. The domain is clustered in the direction of the $y$-axis along the bottom surface and also clustered in the $x$-direction at the area of the morphing bump. The domain was divided into 10 blocks where the cells are nearly orthogonal and quadrilateral. The grid will be updated each time step during the motion of the bump, preserving the body-fitted mesh by applying the Arbitrary Lagrangian-Eulerian (ALE). Four types of boundaries define the borders of the domain: inlet, outlet (right and top after the wedge), non-slip walls (bottom and wedge), and slip wall (top before wedge). Figure 4 shows the boundary and initial conditions over the Schematic of the grid.

A mesh-independent study is essential to ensure the result’s accuracy. The study was implemented over the case of without bump, starting with mesh 1 of size $100\times 45$ cells. Then, high-density meshes (2, 3, and 4) were generated by doubling the number of cells in $x$ and $y$ directions. The time step was controlled with the Courant–Friedrichs–Lewy (CFL) number of $0.2$. Figure. 5 shows the variation of the pressure distribution at height of $y=0.5m$ for different mesh sizes. The results with mesh 3 show a converged solution that does not depend on the mesh quality. This is because the difference between mesh 3 and mesh 4 is negligible. Thus, Mesh 3 of size $400\times 180$ was chosen to implement all the simulations to minimize computational time. The minimum element size in mesh 3 is $2.8mm\times 1.2mm$, and the time step to ensure the CFL number of $0.2$ is $111.1\mu s$.

4.   Results and Discussion

4.1 Shock Structure over a Stationary Bump

Firstly, the benefit of using the bump in the SWBLI problem has to be shown before investigating the transitional effect of the morphing bump. Thus, supersonic flows over a flat plate without and with a stationary bump were simulated to obtain the steady-state solutions of the morphing bump limiting positions, as shown in Figure 6. The benefit of using the bump is indicated by plotting the pressure distribution over the lower boundary for both problems; without and with a stationary bump, as shown in Figure 7.

The existence of the bump created a series of weak expansion fans which in total decreased the pressure dramatically to the inlet value of pressure as shown in Figure 7 within the region of $x\approx(0,0.5)$. Then, the pressure increased at the end of the bump to the original case (clean flat plate) at $x=0.5$, due to the generation of the trailing oblique shock. This decrease of the pressure in this simple geometry (flat plate) case shows that the bump is applicable to control the SWBLI in the airfoil case, where the pressure decreases over the upper surface, and accordingly the lift force increases. This aligns with the results of Mazaheri et al. [12].

4.2 Shock Dynamics over Morphing Bump

The transition phase of the morphing bump was investigated in detail with a constant velocity profile. The motion of the morphing bump occurred during different dimensionless times, $t^{\ast}_{f}=10$, $7.5$, $5$, $3$, $2$, $1.8$, $1$, and $0.1$. This corresponds to a constant local Mach number, $M_{b}$, of $0.004$, $0.00533$, $0.008$, $0.0133$, $0.02$, $0.0222$, $0.04$, and $0.4$ at the mid-point of the morphing bump ($x=0$), respectively, and zero-velocity points at the two edges of the bump. Figure 9 shows the transitional solution of the morphing bump with the slowest constant velocity (the Mach number at $x=0$ is $0.004$ and the final dimensionless time, $t^{\ast}_{f}$ is $10$) by plotting the pressure contours, shown in (I), zoomed picture of the morphing mesh, shown in (II), and entropy, temperature, pressure, and density distributions, shown in (III) and (IV), along the bottom boundary at different dimensionless times, $t^{\ast}$.

There is a spatial lag in the leading and trailing oblique shocks between the transitional solution at different values of constant velocity (different values of the final dimensionless times, $t^{\ast}_{f}$ ), and the stationary condition, as shown in Figure 10. This spatial lag in the shock system from the stationary condition increases when the value of the morphing bump’s constant velocity increases, i.e. the operational time of the morphing bump was reduced. This was concluded also in a rotating wedge problem, made by Margha et al. [27]. The spatial lag was clearly indicated by comparing the pressure, and Mach number distributions at a height of $y=1$, between the stationary condition and the morphing conditions at different $t^{\ast}_{f}$, as shown in Figure 11. Further, the change in the spatial lag can be neglected when the operational time of the morphing bump exceeds the final dimensionless time, $t^{\ast}_{f}=5$.

A comparison between the start state (clean flat plate) and the final state (at the end of the morphing motion) over the bump’s surface at different motion periods was performed by obtaining the flow parameter distributions. Then, the temporal change in the entropy difference distribution with the case of a clean flat plate problem at a steady state over the lower boundary, $\left(\Delta s_{@t=t^{\ast}_{f}}(x)-\Delta s_{@t=0}(x)\right)/c_{v}$, is calculated, as shown in Figure 12. Further, the entropy difference distribution, $\Delta s(x)$, is defined as the difference between the entropy at a certain $x$ location with the inlet condition.

When the leading oblique shock appears due to the morphing bump motion, the pressure and the density over the bump increase. Thus, the temporal change in the entropy difference rises up at the bump’s leading edge. Then, the series of expansion, before and after the incident oblique shock, decrease the pressure and density along the morphing bump. This may cause negative values for the temporal change in the entropy difference over the bump at relatively low morphing speeds. Furthermore, the location of the incident shock moves slightly to the left during the upward motion of the bump. The trailing oblique shock again increases the temporal change in the entropy difference. In addition, there is a deviation between the two entropy difference over the lower boundary; the case of the stationary bump at steady state, and the case of the morphing bump at final states of the motion, see Figure 9 (f) and Figure 10 (a). The deviation happens due to the effect of spatial lag on the shock system, which results from the bump’s motion.

when the morphing bump is moved with a relatively high speed (small $t^{\ast}_{f}$), we lose the gain in the pressure difference ($p_{@t=t^{ast}_{f}}(x)-p_{t=0}(x)$), that is obtained from the series of expansion fans. This is explained as the pressure applied on the morphing bump would increase, which can be considered as a resistant action to the motion of the bump. Further, the compressibility effect increases with the bump’s speed, and consequently the density increases. When the air is compressed with the motion of the bump, the molecular distances decrease, and then the friction between adjacent molecules increases. Thus, the temperature increases with the bump’s motion. Thus, when the morphing bump moves faster, the flow parameters (pressure, density, and temperature) increase, as shown in Figure 12 (a). This is clearly shown in the temporal change in the entropy difference distribution of Figure 12 (b). Hence, the results show that the slowest velocity for the morphing bump, which approximates a quasi-steady flow during the motion, decreases the lag effect.

Despite the existence of the lag effect in the shock system, the slowest tested morphing bump’s speed ($t^{\ast}_{f}=10$) is not the optimal one, due to the deviation in the entropy from the stationary bump at steady state, see Figure 9 (f). This deviation represents the losses in the shock system, resulting from the morphing speed. Thus, comparing the entropy difference for each speed was made with the stationary bump at a steady state, as shown in Figure 13. The results show the suitable motion period, $t^{\ast}_{f}$, for the morphing bump at a free-stream Mach number of 2.9 is 2. The reason is that its entropy difference is very close to that of the stationary bump at a steady state. Furthermore, the lag effect in the system resulting from that morphing speed is within acceptable moderate levels, as shown in Figure 11.

5.   Conclusion

The current research work aimed to investigate the morphing of a bump control element over a flat plate at a free-stream Mach number of $2.9$ and a high Reynolds number of $6.6\times 10^{7}$. Different values of constant velocity were conducted to study the effect of the morphing bump on the shock system with non-dimensional motion periods, $t^{\ast}_{f}$ of $0.1$, $1.0$, $1.8$, $2.0$, $3.0$, $5.0$, $7.5$, $10.0$. Further, the steady state of supersonic flows over a clean flat plate and a flat plate with a stationary bump were conducted to show the beneficial effect of the bump’s existence. Furthermore, a comparison between the dynamic and static cases was achieved. The results showed that a spatial lag in the shock system appears due to the dynamic motion. In addition, the lag effect remarkably increases when the local Mach number at the tip of the morphing bump, $M_{b}$, increases higher than $0.008$. The relatively fast morphing bump would increase the compressibility effect in the near area which may compensate for the beneficial effect of the bump’s existence. Besides, the relatively slow morphing speed results in a deviation in the entropy from the stationary bump case, which is a representation of flow momentum losses. Thus, the suitable speed to morph with is the one that results in neither a remarkable lag effect in the shock system nor high losses in the entropy deviation from the stationary steady-state case. For the case of supersonic flow with $M_{\infty}=2.9$ over a flat plate, the suitable bump’s morphing period, $t^{\ast}_{f}$, was found to be 2. For future work, various velocity profiles with different speed values are recommended to be tested for the morphing bump. This will determine the suitable morphing bump’s velocity profile and speed, that achieves low time-average entropy for the flat plate problem. Then, this configuration of motion will be tested over a transonic airfoil to achieve a high time-averaged lift-to-drag ratio.