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Assessment of d’Alembert’s paradox in panel methods by tangency correction
Engineering Analysis with Boundary Elements 85 (2017) 136–141
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Assessment of d’Alembert’s paradox in panel methods by tangency correction Josip Bašić a,∗, Nastia Degiuli b, Dario Ban a a b
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudera Boskovica 32, Split 21000, Croatia Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 5, Zagreb 10000, Croatia
a r t i c l e Keywords: Potential flow Panel methods Flow around cylinder d’Alembert’s paradox Viscous pressure drag
i n f o
a b s t r a c t The aim of this study is to introduce a correction for panel methods in order to obtain correct viscous pressure drag of a steadily moving body in a viscous fluid. The boundary tangency correction implemented through a surface normal deflection term was added into a panel method implementation. A quadratic programming algorithm was used to solve inverse problems in order to prove the correction well-posedness and to obtain accurate surface pressure distribution, based on the experimental data. The experimental surface pressure distributions were collected from the literature for circular cylinders in steady uniform flow. The solutions of normal deflections were validated and the direct connection of the introduced correction to the boundary layer displacement thickness was found, which can simplify the process of viscous–inviscid coupling between integral boundary layer and panel methods. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In potential flows that assume an ideal fluid without the viscosity, only pressure and inertial forces determine the flow dynamics. D’Alembert showed that due to the symmetry of the pressure field, the drag force is zero for incompressible and inviscid potential flow around a body moving with constant velocity relative to the fluid. Prandtl reasoned that flow fields of real viscous fluids develop very differently near solid boundaries compared to ideal fluids, even for high Reynolds numbers. Inside the boundary layer (BL) formed along solid boundaries, viscous forces balance inertia and pressure gradient forces. In practical cases, most BLs become unstable, so that any small disturbance triggers a transition to the erratically unsteady condition known as the turbulence. Great efforts have been made to explain how such small frictional force in the fluid can have such significant effect on the flow properties. Generally, the actual pressure in the rear of a body is much lower than pressure obtained with the potential flow theory. There are several techniques that simulate the BL by coupling with potential flow codes, e.g. velocity decomposition formulations of the incompressible Navier–Stokes equations [1,2], or integral BL methods solved with empirical closure relations [3]. The coupling of the potential and integral BL solver via pressure and velocity field can be performed by physically shifting boundaries outward in their normal direction using the calculated displacement thickness [4]. Other
∗
transpiration methods “blow” specified velocity from each panel to displace the surface streamlines [5]. In this paper, a novel technique to modify the pressure distribution within a panel method is presented. The technique is based on the treatment of boundary tangency in a way that it does not require any change in panel methods formalism, boundary conditions, nor the geometry, as other methods require. Furthermore, it retains a direct connection to von Kármán’s BL theory through the displacement thickness. The verification of the method is based on experimental data from the literature concerning flows around circular cylinders. Well-posedness is verified by solving inverse problems that connect the numerical method with the experimental data, in order to obtain accurate viscous pressure drag. Viscous–inviscid direct and quasi-simultaneous coupling methods [4] mostly use empirically obtained displacement thickness to achieve coupling between a BL solver and a potential flow solver. The displacement thickness is usually obtained by empirical closure relations for the integral BL equations [3]. The tangency correction modifies the pressure distribution and its connection to the displacement thickness thus allows for straightforward exchange of pressure, velocity and displacement thickness information in the coupling process. The paper is organized as follows. In Section 2, potential flow theory and source-panel method enhancements are introduced. Section 3 presents collected experimental data for flows around a circular cylinder and corresponding inverse problem setup with the panel method in order to yield results approximately equal to the experiments.
Corresponding author. E-mail addresses: [email protected] (J. Bašić), [email protected] (N. Degiuli), [email protected] (D. Ban).
https://doi.org/10.1016/j.enganabound.2017.10.002 Received 11 April 2017; Received in revised form 10 July 2017; Accepted 1 October 2017 0955-7997/© 2017 Elsevier Ltd. All rights reserved.
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Engineering Analysis with Boundary Elements 85 (2017) 136–141
In Section 4, the results of the numerical analysis are discussed. Finally, the conclusions of the conducted study are drawn in Section 5. 2. The methodology 2.1. Potential flow The Laplace equation is obtained from the Navier–Stokes equations by neglecting viscous and nonlinear terms, and by assuming that the flow is irrotational in order to introduce a velocity potential. The flow is therefore restricted to be inviscid, irrotational and linear. Physically, these restrictions prevent the reproduction of important fluid flow behaviours such as the separation, skin-friction drag and transonic shocks. Since the flow is irrotational, the velocity field is curl-free: ∇ × 𝒖 = 0,
Fig. 1. Schematic of panels rotation, the grey and black arrows denote surface normals before and after the deflection angles are applied, respectively.
2.3. The correction for viscous flows
(1)
where u is the fluid velocity field. The curl of the gradient of any scalar field is always null vector, thus a velocity field can be described with a gradient of some scalar function: 𝒖 = ∇𝜙,
The idea is to mathematically impose tangential flow condition expressed with Eq. (5), and physically let the flow not to be strictly tangential to the surface boundary. Rather, the flow follows the outer BL frontier, and after the separation it follows the mean wake direction. The statement is eligible until the flow has fully separated, because the pressure distribution throughout the BL in the direction normal to the surface remains constant. Generally, but mostly pronounced in the wake region, resulting potential will not be zero in a body interior, i.e. the surface boundary will “leak” potential. This is justifiable, since we are not concerned with the inner region of the BL right near the surface, but the mean flow direction and the pressure outside of the BL. This tangency correction is implemented through virtual rotation of panel normals and correspondingly tangents. In two dimensions, scalar panel deflection angle term is introduced, Δ𝛽 i , so that Eq. (7) is modified in the following way: [ ( )] 𝐛𝑖 = −𝑈∞ cos 𝛼∞ − 𝛽𝑖 + Δ𝛽𝑖 , (8)
(2)
where 𝜙 is the scalar potential function. Since the flow is incompressible, it holds that the velocity field is solenoidal: ∇ ⋅ 𝒖 = 0.
(3)
Substituting Eq. (2) into Eq. (3), the Laplace equation is obtained: ∇2 𝜙 = 0.
(4)
The flow is tangential to the boundary and the resulting potential is zero in the body interior, i.e. fluid is neither able to enter or leave a closed surface. This boundary condition (BC) can be represented as a Neumann BC, defined as: 𝒏 ⋅ 𝒖 = 𝒏 ⋅ ∇𝜙 = 𝑢 𝑛 ,
where Δ𝛽 i is the deflection angle of ith panel’s normal, positive in the counter-clockwise direction. Now Eq. (6) is solved with the new r.h.s. described with Eq. (8), and integral equations (see e.g. [6]) are calculated with the newly introduced panel angles, which are schematically shown in Fig. 1.
(5)
where n is the surface normal vector pointing outwards from the body and un is the normal velocity of the body surface. Eq. (5) is the only BC, meaning that the no-slip BC can only be simulated by using extensions in numerical implementations of the potential flow theory.
3. Numerical analysis 2.2. Panel methods
3.1. Flow around a circular cylinder
A two-dimensional body moving with subsonic constant speed in an ideal fluid of infinite extent is considered. The problem is equivalent to a uniform stream in the body-fixed coordinate system. For subsonic freestream flow, the Laplace Eq. (4) has a property that any disturbance is felt everywhere in the flow field, albeit its effect dies rapidly with distance from the disturbance location. The linear nature of potential theory allows the solution of Eq. (4) to be obtained by summing elementary solutions located on the body boundary, i.e. on panels. There are many choices as how to formulate a panel method with singularity solutions [6]. In this study, a source distribution of constant strength 𝜎 i is placed on each ith panel. The problem can simply be represented with a system of linear equations: 𝐀𝝈 = 𝐛,
It can be argued that much of what is known about fluid dynamics, and a great deal of what is still needed to understand and predict, is present in the variety of phenomena generated by the flow around a circular cylinder. This simple body shape has challenged generations of experimentalists and latterly has proved to be an exciting test case for computationalists. It yields all relevant sequences of flow patterns such as flow separation, turbulence transition, reattachment of the flow and further turbulent separation of the BL. The circular cylinder is also chosen as a relevant shape in this study, since it yields much larger discrepancy of the Navier–Stokes and the inviscid solution, compared to streamlined bodies [1]. The characteristics of the flow around a cylinder varies depending upon the relevant Reynolds number, which is given as:
(6) 𝑅𝑒 =
where A is the coefficient influence matrix, and b is the r.h.s. vector given for two-dimensional problems as: ( ) 𝐛𝑖 = −𝑈∞ cos 𝛼∞ − 𝛽𝑖 , 1 ≤ 𝑖 ≤ 𝑁, (7)
𝑈∞ 𝐷 , 𝜈
(9)
where D is the cylinder diameter and 𝜈 is the kinematic viscosity of the fluid. Roshko [7] defined four different flow regimes depending on the characteristics of the flow: subcritical, critical, supercritical and hypercritical. In a subcritical regime (103 < Re < 2 · 105 ), transition to turbulence occurs in the separated shear layers with constant frequency of vortex shedding. In a critical regime (2 · 105 < Re < 5 · 105 ) the drag coefficient decreases and the transition to turbulence is characterized by
where U∞ is the free-stream speed, 𝛼 ∞ is the free-stream direction angle with respect to the x-axis in the counter-clockwise sense, and 𝛽 i is the angle of ith panel’s normal with respect to the x-axis in the counterclockwise sense. 137
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Table 1 List of experiments of flows around a circular cylinder. #
Authors
Re
Regime
Cp
1 2 3 4 5 6 7 8 9 10 11
Schiller & Linke Norberg Norberg Achenbach Achenbach Bursnall & Loftin Tani Achenbach Warschauer & Leene Achenbach Roshko
2.8 · 103 3.9 · 103 8.0 · 103 1.0 · 105 2.6 · 105 4.1 · 105 4.7 · 105 8.5 · 105 1.3 · 106 3.6 · 106 8.6 · 106
Subcritical Subcritical Subcritical Subcritical Transitional Critical Critical Supercritical Supercritical Hypercritical Hypercritical
Fig. 2 Fig. 2 Fig. 2 Fig. 2 Fig. 3 Fig. 3 Fig. 3 Fig. 4 Fig. 4 Fig. 4 Fig. 4
Fig. 3. Pressure coefficient measured along cylinder half-perimeter for transitional and critical regimes.
Fig. 2. Pressure coefficient measured along cylinder half-perimeter for subcritical regime.
the separation with further reattachment of the BL, forming a laminar separation bubble on one side of the cylinder surface. In a supercritical regime (5 · 105 < Re < 2 · 106 ) laminar separation bubbles forms on both sides of the cylinder surface and with probable, but fluctuating vortex shedding. In a hypercritical regime (Re > 2 · 106 ) the drag coefficient is increasing again and the fully turbulent separation of the flow moves upstream, which results in a wider wake. Amongst numerous available experimental data in the literature, eleven assertive experiments taken from [7–13] are included in the analysis and are listed in Table 1. Experimental data are still not available for Re > 107 , and generally there is considerable data disagreement in the range 106 < Re < 107 . Much of the disagreement has been attributed to surface roughness and free-stream turbulence level [14]. The distributions of time-averaged measured pressure coefficient for all flow regimes are shown in Figs. 2–4 for experiments listed in Table 1. The pressure coefficient is a dimensionless number which describes relative pressure in a flow: ( )2 𝑝 (𝒙 ) − 𝑝 ∞ ‖𝒖(𝒙)‖ 𝐶 𝑝 (𝒙 ) = =1− , (10) 1 𝑈∞ 2 𝜌𝑈∞ 2
Fig. 4. Pressure coefficient measured along cylinder half-perimeter for super- and hypercritical regimes.
with LES for subcritical, critical and supercritical regimes are taken from [15–17], and included in Figs. 2–4, respectively. LES can reproduce flows up to hypercritical regime with high fidelity, but requires large amount of computational power and time to prepare and run a simulation.
3.2. Inverse problem An inverse problem is based upon the comparison of numerically obtained pressure distribution along the body surface and the experimental data. Inverse problems are solved for the experiments given in Table 1 on multiple panel-mesh refinement levels, in order to prove solution uniqueness, independent of the discretization. The solution that is obtained in each panel centre is finally interpolated with a cubic Bspline curve. The optimization problem is written as:
where p(x) is the pressure at the point at which the coefficient is being evaluated, p∞ is the static pressure in the free-stream and 𝜌 is the fluid density. The horizontal axis on the figures depicts angular coordinate on the cylinder perimeter, 𝜑, where 𝜑 = 0◦ is located at the front stagnation point, and 𝜑 = 180◦ is located downstream behind the cylinder. This is analogous to an angle between the free-stream direction and the inward normal of the cylinder surface. The status of viscous Navier– Stokes solvers applied on circular cylinders is given in [15–17] where the authors agree that for such bluff bodies more complex solvers such as Large Eddy Simulation (LES) should be preferred over ReynoldsAveraged Navier–Stokes (RANS) solvers. Pressure distributions obtained
‖ ‖2 minimize ‖𝐂𝑝 (𝐁) − 𝐂𝑝,𝑒𝑥𝑝 ‖ ‖ ‖2 𝐁 subject to 𝐁 + 𝐀 ≥ 0 𝐁 + 𝐀 ≤ 𝜋, 138
(11)
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Engineering Analysis with Boundary Elements 85 (2017) 136–141
Fig. 5. Pressure coefficient distribution and corresponding streamlines for experiment #1.
Fig. 7. Pressure coefficient distribution and corresponding streamlines for experiment #11.
Fig. 6. Pressure coefficient distribution and corresponding streamlines for experiment #7.
where B is the vector of Δ𝛽 values located at panel centres, Cp is the vector function that returns pressure coefficients at panel centres, Cp, exp is the vector of experimentally obtained pressure coefficients at panel centres and A is the vector of panels’ angles 𝛽. As seen in Eq. (11), the 𝓁 2 -norm of the difference vector between the numerically obtained and the experiment pressure coefficients is used as a solution discrepancy indicator for the optimization solver. The solver constraint of maximum allowed deflection angle of a panel was set to the value of its angle relative to free-stream, meaning that a panel normal can maximally turn back to free-stream source direction. In addition to different panel-mesh refinement levels, solution uniqueness was verified by random initial [ ] conditions of the vector B, generated in the range −𝜋∕2, 𝜋∕2 . In this study, the Sequential Least Squares Programming (SLSQP) optimization algorithm [18] is used to find solutions to inverse problems, described with Eq. (11). SLSQP is a sequential quadratic programming optimization algorithm, which can be used to solve nonlinear programming problems that minimize a scalar function that is subjected to general equality and inequality constraints and to lower and upper bounds on the variables. The SLSQP solver tries to minimize the norm of the difference vector until the convergence within specified tolerance is achieved.
Fig. 8. Contour plot of panel deflection angle solutions interpolated between Reynolds numbers and their angles.
for experiments #1, #7 and #11, respectively. Fig. 5 presents a subcritical flow characterized with early laminar separation (see Fig. 2) and thus very wide wake. On the other hand, the solution for a hypercritical flow is presented in Fig. 6, which is characterized with late turbulent separation (see Fig. 3) and a thin wake. Fig. 7 presents a hypercritical flow with medium-late separation (see Fig. 4), and medium-wide wake. It can be observed that the streamlines appropriately follow separated flow that reduced the curvature of the streamlines. In addition, the calculated flow respects realistic wake size and mean flow speed (rendered with streamlines thickness). The pressure obtained by including solved deflection angle distributions in Eq. (8) converge to experimental pressure plotted in Figs. 2–4, as the panel count of the cylinder is increasing. Fig. 8 renders interpolated contour plot with the solutions of the deflection angle for the considered cylinder experiments. For subcritical flows, the deflection angles start to persistently change at around 𝜑 = 70◦ . For hypercritical flows, the change starts at around 𝜑 = 100◦ , and for critical flows the change is delayed to around 𝜑 = 125◦ . The solutions rendered in Fig. 8 show that realistic pressure distributions are mostly controlled with the tangency correction in the downstream part of the cylinder. The solutions of the deflection angle follow the trend of accumulated discrepancy between potential and real flow pressure distribution, shown in Figs. 2–4. Interestingly, they are never discontinuous near separation
4. Results and discussion 4.1. Solution of the inverse problem The inverse problems are solved for the experiments presented in Section 3. Contour plots of the solved pressure coefficient and corresponding flow streamlines around the cylinder are shown in Figs. 5–7 139
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Engineering Analysis with Boundary Elements 85 (2017) 136–141
points, i.e. deflection angle distributions and obtained pressure distributions are smooth functions. On the other hand, the Laplace equation by its nature gives smooth solutions even for the rough boundary data, and thus even separated flows have smooth distribution of normals deflection. In the following text, these observations will be augmented with the physical context of the tangency correction. 4.2. The tangency and the boundary layer Assume a fluid particle that travels from the front stagnation point downstream. It does not truly follow the solid boundary. Rather, its timeaveraged path is displaced from the boundary. Assume that the particle is followed over the displaced path, i.e. that the tangens of the deflection angle is integrated over the path from the stagnation point. The results of the integration is effectively the displacement thickness, which is defined as the distance by which an ideal flow streamline is displaced to obtain the flow rate of the viscous flow. This can formulated in the following way: | | 𝑥 𝛿 ⋆ (𝑥) ≈ || tan [Δ𝛽(𝑥̃ )]d𝑥̃ ||, (12) ∫ | | 0 where it holds that the displacement thickness at upstream stagnation point is negligible, since according to Thwaites √ [6] the initial momentum thickness for a cylinder equals 𝜃(0) = 0.27𝑅∕ 𝑅𝑒. In conclusion, Eq. (12) is the direct connection between the deflection angle and the displacement thickness. For a known velocity distribution along the surface normal, u(y), the displacement thickness can classically be expressed as: 𝛿 ⋆ (𝑥 ) =
∞
∫0
( ) 𝑢 (𝑦 ) 1− d𝑦, 𝑈
Fig. 9. Displacement and momentum thicknesses, and skin friction coefficient for experiment #10.
After the inverse problem is solved, the displacement and momentum thicknesses are obtained using Eqs. (12) and (16), respectively, and the results up to complete flow separation are shown in Fig. 9. The figure shows that the friction coefficient reaches negative value at 𝜑 = 78◦ where the flow separation starts, which is characterized by rapid growth of the displacement thickness. The comparison of the growth trends obtained with Eq. (16) and with LES shows good agreement in the vicinity of the flow separation. Moreover, the comparison of the momentum thickness distributions obtained with Eq. (16) and referent laminar solution obtained with Eq. (15) shows good agreement. It can be stated that the BL properties are well reproduced and that connection between the tangency correction and the displacement thickness described with Eq. (12) holds.
(13)
where y is the distance from the surface along the normal and U is the potential flow speed at the edge of the BL. Von Kármán proposed the famous momentum integral equation, obtained by integrating the momentum equation across the BL while keeping relevant terms, which is for two-dimensional steady incompressible flow given as: 𝐶𝑓 ( )𝑈′ 𝜃 ′ + 2𝜃 + 𝛿 ⋆ = , (14) 𝑈 2 where 𝜃 is the momentum thickness, Cf is the skin friction coefficient defined as 𝜏 w /(0.5𝜌U2 ), 𝜏 w is the wall stress and the superscript ′ denotes the derivative with respect to x. The unsteady version of Eq. (14) includes the term d(U𝛿 ⋆ )/dt on the l.h.s. Eq. (14) also holds along a curved wall provided that the radius of curvature is much greater than the BL thickness, and is valid for all types of fluid flow: laminar, turbulent or transitional. Of course, the BL properties are functions of curved wall location x. In order to verify that Eq. (12) holds, BL properties of experiment #4 are compared to the numerical solution. Cp (x) and Cf (x) distribution of experiment #4 are taken from [10]. Referent displacement thickness values near the flow separation are obtained by employing Eq. (13) and the velocity distribution taken from [19], where a LES simulation of subcritical flow around a cylinder was performed for 𝑅𝑒 = 1.26 ⋅ 105 . Additionally, referent momentum thickness distribution is calculated with the empirical Thwaites’ integral formulation [6]: 𝜃(𝑥)2 ≈
5. Conclusions A novel treatment of the boundary condition for panel methods is introduced. The treatment takes the effects of the boundary layer displacement thickness into account, with the purpose that a panel method can approximately reconstruct pressure distribution of viscous flow around a moving body. The treatment, called the tangency correction, is implemented through the virtual rotation of panel normals and correspondingly tangents, without any change in the formalism of panel methods. An outcome besides the accurate pressure distribution, is that the computed flow really follows the outer boundary layer frontier, and after the separation it follows the mean wake direction. The verification of the tangency correction well-posedness was done by solving inverse problems that connect the numerical method with the experimental data from the literature concerning flows around circular cylinders. The physical context of the tangency correction in correlation with the displacement thickness was deduced and validated. As the methods that couple integral BL equations with a panel method are mostly established by exchanging the pressure/velocity and the displacement thickness information, the introduced tangency correction can be used in viscous–inviscid coupling process. The viscous–inviscid coupling is applicable on threedimensional problems, similarly as in [20], in which case the tangency correction can be applied marching on a streamline downstream from the stagnation point. Also, unsteady term in the integral BL equations can be taken into account for unsteady flows [21].
𝑥
0.45𝜈 𝑈 (𝑥̃ )5 d𝑥̃ , 𝑈 (𝑥)6 ∫0
(15)
which holds for laminar boundary layers. If a Cf (x) distribution is known e.g. from an experiment, then the linear first-order ordinary differential Eq. (14) is solved for 𝜃. The solution is given with the following expression: ] [ 𝑥 𝐼 (𝑥) 𝐶 (𝑥 𝛿 ⋆ (𝑥̃ )𝐶𝑝′ (𝑥̃ ) 𝑝 𝑓 ̃) 𝜃(𝑥) = + ( (16) ) d𝑥̃ , ∫0 𝐼𝑝 (𝑥̃ ) 2 2 1 − 𝐶𝑝 (𝑥̃ ) where Ip (x) is a function defined as: ( ) 𝑥 𝐶𝑝′ (𝑥̃ ) 𝐼𝑝 (𝑥) = exp d𝑥̃ . ∫0 1 − 𝐶𝑝 (𝑥̃ )
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Assessment of d’Alembert’s paradox in panel methods by tangency correction Josip Bašić a,∗, Nastia Degiuli b, Dario Ban a a b
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudera Boskovica 32, Split 21000, Croatia Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 5, Zagreb 10000, Croatia
a r t i c l e Keywords: Potential flow Panel methods Flow around cylinder d’Alembert’s paradox Viscous pressure drag
i n f o
a b s t r a c t The aim of this study is to introduce a correction for panel methods in order to obtain correct viscous pressure drag of a steadily moving body in a viscous fluid. The boundary tangency correction implemented through a surface normal deflection term was added into a panel method implementation. A quadratic programming algorithm was used to solve inverse problems in order to prove the correction well-posedness and to obtain accurate surface pressure distribution, based on the experimental data. The experimental surface pressure distributions were collected from the literature for circular cylinders in steady uniform flow. The solutions of normal deflections were validated and the direct connection of the introduced correction to the boundary layer displacement thickness was found, which can simplify the process of viscous–inviscid coupling between integral boundary layer and panel methods. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In potential flows that assume an ideal fluid without the viscosity, only pressure and inertial forces determine the flow dynamics. D’Alembert showed that due to the symmetry of the pressure field, the drag force is zero for incompressible and inviscid potential flow around a body moving with constant velocity relative to the fluid. Prandtl reasoned that flow fields of real viscous fluids develop very differently near solid boundaries compared to ideal fluids, even for high Reynolds numbers. Inside the boundary layer (BL) formed along solid boundaries, viscous forces balance inertia and pressure gradient forces. In practical cases, most BLs become unstable, so that any small disturbance triggers a transition to the erratically unsteady condition known as the turbulence. Great efforts have been made to explain how such small frictional force in the fluid can have such significant effect on the flow properties. Generally, the actual pressure in the rear of a body is much lower than pressure obtained with the potential flow theory. There are several techniques that simulate the BL by coupling with potential flow codes, e.g. velocity decomposition formulations of the incompressible Navier–Stokes equations [1,2], or integral BL methods solved with empirical closure relations [3]. The coupling of the potential and integral BL solver via pressure and velocity field can be performed by physically shifting boundaries outward in their normal direction using the calculated displacement thickness [4]. Other
∗
transpiration methods “blow” specified velocity from each panel to displace the surface streamlines [5]. In this paper, a novel technique to modify the pressure distribution within a panel method is presented. The technique is based on the treatment of boundary tangency in a way that it does not require any change in panel methods formalism, boundary conditions, nor the geometry, as other methods require. Furthermore, it retains a direct connection to von Kármán’s BL theory through the displacement thickness. The verification of the method is based on experimental data from the literature concerning flows around circular cylinders. Well-posedness is verified by solving inverse problems that connect the numerical method with the experimental data, in order to obtain accurate viscous pressure drag. Viscous–inviscid direct and quasi-simultaneous coupling methods [4] mostly use empirically obtained displacement thickness to achieve coupling between a BL solver and a potential flow solver. The displacement thickness is usually obtained by empirical closure relations for the integral BL equations [3]. The tangency correction modifies the pressure distribution and its connection to the displacement thickness thus allows for straightforward exchange of pressure, velocity and displacement thickness information in the coupling process. The paper is organized as follows. In Section 2, potential flow theory and source-panel method enhancements are introduced. Section 3 presents collected experimental data for flows around a circular cylinder and corresponding inverse problem setup with the panel method in order to yield results approximately equal to the experiments.
Corresponding author. E-mail addresses: [email protected] (J. Bašić), [email protected] (N. Degiuli), [email protected] (D. Ban).
https://doi.org/10.1016/j.enganabound.2017.10.002 Received 11 April 2017; Received in revised form 10 July 2017; Accepted 1 October 2017 0955-7997/© 2017 Elsevier Ltd. All rights reserved.
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In Section 4, the results of the numerical analysis are discussed. Finally, the conclusions of the conducted study are drawn in Section 5. 2. The methodology 2.1. Potential flow The Laplace equation is obtained from the Navier–Stokes equations by neglecting viscous and nonlinear terms, and by assuming that the flow is irrotational in order to introduce a velocity potential. The flow is therefore restricted to be inviscid, irrotational and linear. Physically, these restrictions prevent the reproduction of important fluid flow behaviours such as the separation, skin-friction drag and transonic shocks. Since the flow is irrotational, the velocity field is curl-free: ∇ × 𝒖 = 0,
Fig. 1. Schematic of panels rotation, the grey and black arrows denote surface normals before and after the deflection angles are applied, respectively.
2.3. The correction for viscous flows
(1)
where u is the fluid velocity field. The curl of the gradient of any scalar field is always null vector, thus a velocity field can be described with a gradient of some scalar function: 𝒖 = ∇𝜙,
The idea is to mathematically impose tangential flow condition expressed with Eq. (5), and physically let the flow not to be strictly tangential to the surface boundary. Rather, the flow follows the outer BL frontier, and after the separation it follows the mean wake direction. The statement is eligible until the flow has fully separated, because the pressure distribution throughout the BL in the direction normal to the surface remains constant. Generally, but mostly pronounced in the wake region, resulting potential will not be zero in a body interior, i.e. the surface boundary will “leak” potential. This is justifiable, since we are not concerned with the inner region of the BL right near the surface, but the mean flow direction and the pressure outside of the BL. This tangency correction is implemented through virtual rotation of panel normals and correspondingly tangents. In two dimensions, scalar panel deflection angle term is introduced, Δ𝛽 i , so that Eq. (7) is modified in the following way: [ ( )] 𝐛𝑖 = −𝑈∞ cos 𝛼∞ − 𝛽𝑖 + Δ𝛽𝑖 , (8)
(2)
where 𝜙 is the scalar potential function. Since the flow is incompressible, it holds that the velocity field is solenoidal: ∇ ⋅ 𝒖 = 0.
(3)
Substituting Eq. (2) into Eq. (3), the Laplace equation is obtained: ∇2 𝜙 = 0.
(4)
The flow is tangential to the boundary and the resulting potential is zero in the body interior, i.e. fluid is neither able to enter or leave a closed surface. This boundary condition (BC) can be represented as a Neumann BC, defined as: 𝒏 ⋅ 𝒖 = 𝒏 ⋅ ∇𝜙 = 𝑢 𝑛 ,
where Δ𝛽 i is the deflection angle of ith panel’s normal, positive in the counter-clockwise direction. Now Eq. (6) is solved with the new r.h.s. described with Eq. (8), and integral equations (see e.g. [6]) are calculated with the newly introduced panel angles, which are schematically shown in Fig. 1.
(5)
where n is the surface normal vector pointing outwards from the body and un is the normal velocity of the body surface. Eq. (5) is the only BC, meaning that the no-slip BC can only be simulated by using extensions in numerical implementations of the potential flow theory.
3. Numerical analysis 2.2. Panel methods
3.1. Flow around a circular cylinder
A two-dimensional body moving with subsonic constant speed in an ideal fluid of infinite extent is considered. The problem is equivalent to a uniform stream in the body-fixed coordinate system. For subsonic freestream flow, the Laplace Eq. (4) has a property that any disturbance is felt everywhere in the flow field, albeit its effect dies rapidly with distance from the disturbance location. The linear nature of potential theory allows the solution of Eq. (4) to be obtained by summing elementary solutions located on the body boundary, i.e. on panels. There are many choices as how to formulate a panel method with singularity solutions [6]. In this study, a source distribution of constant strength 𝜎 i is placed on each ith panel. The problem can simply be represented with a system of linear equations: 𝐀𝝈 = 𝐛,
It can be argued that much of what is known about fluid dynamics, and a great deal of what is still needed to understand and predict, is present in the variety of phenomena generated by the flow around a circular cylinder. This simple body shape has challenged generations of experimentalists and latterly has proved to be an exciting test case for computationalists. It yields all relevant sequences of flow patterns such as flow separation, turbulence transition, reattachment of the flow and further turbulent separation of the BL. The circular cylinder is also chosen as a relevant shape in this study, since it yields much larger discrepancy of the Navier–Stokes and the inviscid solution, compared to streamlined bodies [1]. The characteristics of the flow around a cylinder varies depending upon the relevant Reynolds number, which is given as:
(6) 𝑅𝑒 =
where A is the coefficient influence matrix, and b is the r.h.s. vector given for two-dimensional problems as: ( ) 𝐛𝑖 = −𝑈∞ cos 𝛼∞ − 𝛽𝑖 , 1 ≤ 𝑖 ≤ 𝑁, (7)
𝑈∞ 𝐷 , 𝜈
(9)
where D is the cylinder diameter and 𝜈 is the kinematic viscosity of the fluid. Roshko [7] defined four different flow regimes depending on the characteristics of the flow: subcritical, critical, supercritical and hypercritical. In a subcritical regime (103 < Re < 2 · 105 ), transition to turbulence occurs in the separated shear layers with constant frequency of vortex shedding. In a critical regime (2 · 105 < Re < 5 · 105 ) the drag coefficient decreases and the transition to turbulence is characterized by
where U∞ is the free-stream speed, 𝛼 ∞ is the free-stream direction angle with respect to the x-axis in the counter-clockwise sense, and 𝛽 i is the angle of ith panel’s normal with respect to the x-axis in the counterclockwise sense. 137
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Table 1 List of experiments of flows around a circular cylinder. #
Authors
Re
Regime
Cp
1 2 3 4 5 6 7 8 9 10 11
Schiller & Linke Norberg Norberg Achenbach Achenbach Bursnall & Loftin Tani Achenbach Warschauer & Leene Achenbach Roshko
2.8 · 103 3.9 · 103 8.0 · 103 1.0 · 105 2.6 · 105 4.1 · 105 4.7 · 105 8.5 · 105 1.3 · 106 3.6 · 106 8.6 · 106
Subcritical Subcritical Subcritical Subcritical Transitional Critical Critical Supercritical Supercritical Hypercritical Hypercritical
Fig. 2 Fig. 2 Fig. 2 Fig. 2 Fig. 3 Fig. 3 Fig. 3 Fig. 4 Fig. 4 Fig. 4 Fig. 4
Fig. 3. Pressure coefficient measured along cylinder half-perimeter for transitional and critical regimes.
Fig. 2. Pressure coefficient measured along cylinder half-perimeter for subcritical regime.
the separation with further reattachment of the BL, forming a laminar separation bubble on one side of the cylinder surface. In a supercritical regime (5 · 105 < Re < 2 · 106 ) laminar separation bubbles forms on both sides of the cylinder surface and with probable, but fluctuating vortex shedding. In a hypercritical regime (Re > 2 · 106 ) the drag coefficient is increasing again and the fully turbulent separation of the flow moves upstream, which results in a wider wake. Amongst numerous available experimental data in the literature, eleven assertive experiments taken from [7–13] are included in the analysis and are listed in Table 1. Experimental data are still not available for Re > 107 , and generally there is considerable data disagreement in the range 106 < Re < 107 . Much of the disagreement has been attributed to surface roughness and free-stream turbulence level [14]. The distributions of time-averaged measured pressure coefficient for all flow regimes are shown in Figs. 2–4 for experiments listed in Table 1. The pressure coefficient is a dimensionless number which describes relative pressure in a flow: ( )2 𝑝 (𝒙 ) − 𝑝 ∞ ‖𝒖(𝒙)‖ 𝐶 𝑝 (𝒙 ) = =1− , (10) 1 𝑈∞ 2 𝜌𝑈∞ 2
Fig. 4. Pressure coefficient measured along cylinder half-perimeter for super- and hypercritical regimes.
with LES for subcritical, critical and supercritical regimes are taken from [15–17], and included in Figs. 2–4, respectively. LES can reproduce flows up to hypercritical regime with high fidelity, but requires large amount of computational power and time to prepare and run a simulation.
3.2. Inverse problem An inverse problem is based upon the comparison of numerically obtained pressure distribution along the body surface and the experimental data. Inverse problems are solved for the experiments given in Table 1 on multiple panel-mesh refinement levels, in order to prove solution uniqueness, independent of the discretization. The solution that is obtained in each panel centre is finally interpolated with a cubic Bspline curve. The optimization problem is written as:
where p(x) is the pressure at the point at which the coefficient is being evaluated, p∞ is the static pressure in the free-stream and 𝜌 is the fluid density. The horizontal axis on the figures depicts angular coordinate on the cylinder perimeter, 𝜑, where 𝜑 = 0◦ is located at the front stagnation point, and 𝜑 = 180◦ is located downstream behind the cylinder. This is analogous to an angle between the free-stream direction and the inward normal of the cylinder surface. The status of viscous Navier– Stokes solvers applied on circular cylinders is given in [15–17] where the authors agree that for such bluff bodies more complex solvers such as Large Eddy Simulation (LES) should be preferred over ReynoldsAveraged Navier–Stokes (RANS) solvers. Pressure distributions obtained
‖ ‖2 minimize ‖𝐂𝑝 (𝐁) − 𝐂𝑝,𝑒𝑥𝑝 ‖ ‖ ‖2 𝐁 subject to 𝐁 + 𝐀 ≥ 0 𝐁 + 𝐀 ≤ 𝜋, 138
(11)
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Fig. 5. Pressure coefficient distribution and corresponding streamlines for experiment #1.
Fig. 7. Pressure coefficient distribution and corresponding streamlines for experiment #11.
Fig. 6. Pressure coefficient distribution and corresponding streamlines for experiment #7.
where B is the vector of Δ𝛽 values located at panel centres, Cp is the vector function that returns pressure coefficients at panel centres, Cp, exp is the vector of experimentally obtained pressure coefficients at panel centres and A is the vector of panels’ angles 𝛽. As seen in Eq. (11), the 𝓁 2 -norm of the difference vector between the numerically obtained and the experiment pressure coefficients is used as a solution discrepancy indicator for the optimization solver. The solver constraint of maximum allowed deflection angle of a panel was set to the value of its angle relative to free-stream, meaning that a panel normal can maximally turn back to free-stream source direction. In addition to different panel-mesh refinement levels, solution uniqueness was verified by random initial [ ] conditions of the vector B, generated in the range −𝜋∕2, 𝜋∕2 . In this study, the Sequential Least Squares Programming (SLSQP) optimization algorithm [18] is used to find solutions to inverse problems, described with Eq. (11). SLSQP is a sequential quadratic programming optimization algorithm, which can be used to solve nonlinear programming problems that minimize a scalar function that is subjected to general equality and inequality constraints and to lower and upper bounds on the variables. The SLSQP solver tries to minimize the norm of the difference vector until the convergence within specified tolerance is achieved.
Fig. 8. Contour plot of panel deflection angle solutions interpolated between Reynolds numbers and their angles.
for experiments #1, #7 and #11, respectively. Fig. 5 presents a subcritical flow characterized with early laminar separation (see Fig. 2) and thus very wide wake. On the other hand, the solution for a hypercritical flow is presented in Fig. 6, which is characterized with late turbulent separation (see Fig. 3) and a thin wake. Fig. 7 presents a hypercritical flow with medium-late separation (see Fig. 4), and medium-wide wake. It can be observed that the streamlines appropriately follow separated flow that reduced the curvature of the streamlines. In addition, the calculated flow respects realistic wake size and mean flow speed (rendered with streamlines thickness). The pressure obtained by including solved deflection angle distributions in Eq. (8) converge to experimental pressure plotted in Figs. 2–4, as the panel count of the cylinder is increasing. Fig. 8 renders interpolated contour plot with the solutions of the deflection angle for the considered cylinder experiments. For subcritical flows, the deflection angles start to persistently change at around 𝜑 = 70◦ . For hypercritical flows, the change starts at around 𝜑 = 100◦ , and for critical flows the change is delayed to around 𝜑 = 125◦ . The solutions rendered in Fig. 8 show that realistic pressure distributions are mostly controlled with the tangency correction in the downstream part of the cylinder. The solutions of the deflection angle follow the trend of accumulated discrepancy between potential and real flow pressure distribution, shown in Figs. 2–4. Interestingly, they are never discontinuous near separation
4. Results and discussion 4.1. Solution of the inverse problem The inverse problems are solved for the experiments presented in Section 3. Contour plots of the solved pressure coefficient and corresponding flow streamlines around the cylinder are shown in Figs. 5–7 139
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points, i.e. deflection angle distributions and obtained pressure distributions are smooth functions. On the other hand, the Laplace equation by its nature gives smooth solutions even for the rough boundary data, and thus even separated flows have smooth distribution of normals deflection. In the following text, these observations will be augmented with the physical context of the tangency correction. 4.2. The tangency and the boundary layer Assume a fluid particle that travels from the front stagnation point downstream. It does not truly follow the solid boundary. Rather, its timeaveraged path is displaced from the boundary. Assume that the particle is followed over the displaced path, i.e. that the tangens of the deflection angle is integrated over the path from the stagnation point. The results of the integration is effectively the displacement thickness, which is defined as the distance by which an ideal flow streamline is displaced to obtain the flow rate of the viscous flow. This can formulated in the following way: | | 𝑥 𝛿 ⋆ (𝑥) ≈ || tan [Δ𝛽(𝑥̃ )]d𝑥̃ ||, (12) ∫ | | 0 where it holds that the displacement thickness at upstream stagnation point is negligible, since according to Thwaites √ [6] the initial momentum thickness for a cylinder equals 𝜃(0) = 0.27𝑅∕ 𝑅𝑒. In conclusion, Eq. (12) is the direct connection between the deflection angle and the displacement thickness. For a known velocity distribution along the surface normal, u(y), the displacement thickness can classically be expressed as: 𝛿 ⋆ (𝑥 ) =
∞
∫0
( ) 𝑢 (𝑦 ) 1− d𝑦, 𝑈
Fig. 9. Displacement and momentum thicknesses, and skin friction coefficient for experiment #10.
After the inverse problem is solved, the displacement and momentum thicknesses are obtained using Eqs. (12) and (16), respectively, and the results up to complete flow separation are shown in Fig. 9. The figure shows that the friction coefficient reaches negative value at 𝜑 = 78◦ where the flow separation starts, which is characterized by rapid growth of the displacement thickness. The comparison of the growth trends obtained with Eq. (16) and with LES shows good agreement in the vicinity of the flow separation. Moreover, the comparison of the momentum thickness distributions obtained with Eq. (16) and referent laminar solution obtained with Eq. (15) shows good agreement. It can be stated that the BL properties are well reproduced and that connection between the tangency correction and the displacement thickness described with Eq. (12) holds.
(13)
where y is the distance from the surface along the normal and U is the potential flow speed at the edge of the BL. Von Kármán proposed the famous momentum integral equation, obtained by integrating the momentum equation across the BL while keeping relevant terms, which is for two-dimensional steady incompressible flow given as: 𝐶𝑓 ( )𝑈′ 𝜃 ′ + 2𝜃 + 𝛿 ⋆ = , (14) 𝑈 2 where 𝜃 is the momentum thickness, Cf is the skin friction coefficient defined as 𝜏 w /(0.5𝜌U2 ), 𝜏 w is the wall stress and the superscript ′ denotes the derivative with respect to x. The unsteady version of Eq. (14) includes the term d(U𝛿 ⋆ )/dt on the l.h.s. Eq. (14) also holds along a curved wall provided that the radius of curvature is much greater than the BL thickness, and is valid for all types of fluid flow: laminar, turbulent or transitional. Of course, the BL properties are functions of curved wall location x. In order to verify that Eq. (12) holds, BL properties of experiment #4 are compared to the numerical solution. Cp (x) and Cf (x) distribution of experiment #4 are taken from [10]. Referent displacement thickness values near the flow separation are obtained by employing Eq. (13) and the velocity distribution taken from [19], where a LES simulation of subcritical flow around a cylinder was performed for 𝑅𝑒 = 1.26 ⋅ 105 . Additionally, referent momentum thickness distribution is calculated with the empirical Thwaites’ integral formulation [6]: 𝜃(𝑥)2 ≈
5. Conclusions A novel treatment of the boundary condition for panel methods is introduced. The treatment takes the effects of the boundary layer displacement thickness into account, with the purpose that a panel method can approximately reconstruct pressure distribution of viscous flow around a moving body. The treatment, called the tangency correction, is implemented through the virtual rotation of panel normals and correspondingly tangents, without any change in the formalism of panel methods. An outcome besides the accurate pressure distribution, is that the computed flow really follows the outer boundary layer frontier, and after the separation it follows the mean wake direction. The verification of the tangency correction well-posedness was done by solving inverse problems that connect the numerical method with the experimental data from the literature concerning flows around circular cylinders. The physical context of the tangency correction in correlation with the displacement thickness was deduced and validated. As the methods that couple integral BL equations with a panel method are mostly established by exchanging the pressure/velocity and the displacement thickness information, the introduced tangency correction can be used in viscous–inviscid coupling process. The viscous–inviscid coupling is applicable on threedimensional problems, similarly as in [20], in which case the tangency correction can be applied marching on a streamline downstream from the stagnation point. Also, unsteady term in the integral BL equations can be taken into account for unsteady flows [21].
𝑥
0.45𝜈 𝑈 (𝑥̃ )5 d𝑥̃ , 𝑈 (𝑥)6 ∫0
(15)
which holds for laminar boundary layers. If a Cf (x) distribution is known e.g. from an experiment, then the linear first-order ordinary differential Eq. (14) is solved for 𝜃. The solution is given with the following expression: ] [ 𝑥 𝐼 (𝑥) 𝐶 (𝑥 𝛿 ⋆ (𝑥̃ )𝐶𝑝′ (𝑥̃ ) 𝑝 𝑓 ̃) 𝜃(𝑥) = + ( (16) ) d𝑥̃ , ∫0 𝐼𝑝 (𝑥̃ ) 2 2 1 − 𝐶𝑝 (𝑥̃ ) where Ip (x) is a function defined as: ( ) 𝑥 𝐶𝑝′ (𝑥̃ ) 𝐼𝑝 (𝑥) = exp d𝑥̃ . ∫0 1 − 𝐶𝑝 (𝑥̃ )
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