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Mach reflection in steady supersonic flow considering wedge boundary-layer correction
CJA 1401 16 October 2019 Chinese Journal of Aeronautics, (2019), xxx(xx): xxx–xxx
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Mach reflection in steady supersonic flow considering wedge boundary-layer correction
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Zijun CHEN a, Chenyuan BAI a,b, Ziniu WU a,*
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a b
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Institute of Aeroengine, Tsinghua University, Beijing 100084, China
Received 14 December 2018; revised 2 January 2019; accepted 26 February 2019
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KEYWORDS
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Boundary-layer correction; Mach reflection; Shock waves; Supersonic flow; Viscous effect
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Abstract Mach reflection in steady supersonic flow is an important phenomenon having received extensive studies, among which simplified theoretical models to predict the size of Mach stem and other flow structure are of particular interest. Past efforts for such models were based on inviscid assumption while in real cases the flow is viscous. Here in this paper we consider the influence of wedge boundary layer on the Mach stem height. This is done by including a simplified boundary layer model into a recently published inviscid model. In this viscous model, the wedge angle and the trailing edge height, which control the Mach stem height, are replaced by their equivalent ones accounting for the displacement effect of the wedge boundary layer, with the boundary layer assumed to be laminar or fully turbulent. This viscous model is shown to compare well with numerical results by computational fluid dynamics and gives a Mach stem height as function of the Reynolds number and Mach number. It is shown that due to the viscous effect, the Mach stem height is increased, through increasing the effective wedge angle. Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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Shock reflection in steady supersonic flow is a well-studied phenomenon.1 As shown in Fig. 1,2 a wedge of length w and angle hw induces an incident shock wave (i), which reflects from the reflecting surface. When the wedge angle is large enough or
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* Corresponding author. E-mail address: [email protected] (Z. WU). Peer review under responsibility of Editorial Committee of CJA.
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the inflow Mach number Ma0 is small enough, the reflected shock wave detaches and a special reflection configuration, called irregular reflection or Mach reflection (since this was first observed by Mach3), occurs. Mach reflection has a three-shock structure. A triple shock point (T) connects these three shock waves, including the incident shock wave (i), the reflected shock wave (r) and a strong shock wave called Mach stem (m). A slipline (s), generated from the triple point, separates the flows downstream of the reflected shock wave and Mach stem and across this slipline the pressure is balanced. The slipline and the reflecting surface enclose a flow region that can be regarded as quasi-onedimensional.4–7 The transmitted expansion waves from the trailing-edge (R) intersect this slipline and reduces the pres-
https://doi.org/10.1016/j.cja.2019.09.007 1000-9361 Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: CHEN Z et al. Mach reflection in steady supersonic flow considering wedge boundary-layer correction, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.09.007
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Fig. 1 Mach reflection configuration following Bai and Wu.2 The reflection of the incident shock wave (i) produces a reflected shock wave (r) composed of a free segment (TF), an interacting segment (FK) and another straight segment (r’), a strong shock wave called Mach stem (m), and a slipline (s) (composed of a free segment TB and an interactive segment BE). Secondary Mach waves are generated over the slipline. A sonic throat exists in the quasi-one-dimensional flow region below the slipline.
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sure, so that the quasi-one-dimensional region or duct has a sonic throat. The three shock waves and the slipline separate the flow into four regions, the solutions of which in the vicinity of the triple point can be found using the three-shock theory of von Neumann.8 The mechanism by which the size of the Mach stem is determined has been a long-standing problem. This problem was initially raised by Courant and Friedrichs9 (see also Liepmann and Roshko10) and about forty years later it was still regarded as unsolved by Emanuel,11 see also Ben-Dor and Takayama.12 Azevedo and Liu4 proposed for the first time a physical model for predicting the Mach stem height. In this model, the sonic throat is assumed to occur where the leading Mach wave of the expansion fan intersects the slip stream and the flow below the slipline is treated using the isentropic quasi-onedimensional ideal gas flow theory. Li and Ben-Dor13 allowed the sonic throat to occur further downstream and assumed that the Mach stem (m), the reflected shock wave (r) and the slipline (s) are slightly curved to be modelled by second order polynomials. Mouton and Hornung6 constructed a model for the growth rate of the Mach stem height during the transition and obtained the Mach stem height as the transition is finished. They assumed that the Mach stem, the reflected shock wave, the slipline and the Mach waves to be straight lines. Gao and Wu7 observed from numerical simulation that secondary Mach waves are generated over slipline to balance the pressure decrease below the slipline and built a characteristic theory to account for the influence of these secondary waves. Bai and Wu2 further derived analytical expressions for the shape of slipline and reflected shock wave and found that the slipline changes slope at the turning point (B) where the leading Mach wave of the expansion fan intersects the slip stream. They then believed that a turning point wave, in the form of a weak shock wave, is generated from this turning point. Recently, Schmisseur and Gaitonde14 performed a numerical study of the Mach stem height taking into account the effect of viscosity. Mach stem heights from inviscid and viscous flow calculations using Computational Fluid Dynamics (CFD) are compared and it was found that viscosity increases the Mach stem height. They pointed out that the increase of
the Mach stem height is due to the increased displacement effect of the wedge boundary-layer and suggested that further work is essential to confirm this hypothesis. In both experimental and numerical studies for the Mach stem height in viscous flow conditions, one often uses the shock angle as input condition, making the influence of viscosity unclear. The boundary layer developed over the lower surface of the wedge not only increases the effective wedge angle and but also reduces the effective height of the wedge trailing edge. Measuring the shock angle automatically includes the influence of the boundary layer in increasing the shock angle, but the effect of the reduced effective trailing-edge height is not considered. Though the displacement thickness of boundary layer is small so that the increase of the effective wedge angle and the decrease of the effective trailing-edge height may be small, its influence on Mach stem may be large since the dependence of the Mach stem height on the geometrical parameters is very sensitive, especially when the Mach stem height is small.13 The purpose of this paper is therefore to built a viscous Mach reflection model for Mach stem height, to account for the influence of the boundary layer developed over the lower surface of the wedge. This influence is expected to be important when the Reynolds number is low (since the displacement thickness of the boundary layer increases for decreasing Reynolds number) or when the Mach number is large (since aerodynamic heating for large Mach number increases the boundary layer thickness15). In the present paper, the reflecting surface is considered as an inviscid one (as in the case of the symmetrical plane of Mach reflection by intersection of two shock waves from opposite sides). For viscous reflecting surface as in the case of a solid wall, the problem is complicated by shock wave/boundary layer interaction.16 In Section 2.1, the viscous model based on the boundary correction to the inviscid model of Bai and Wu2 is introduced. This model gives the displacement wedge angle (Dhw ) and equivalent relative trailing-edge height (Dðg=wÞ) for modification. For details, the inviscid model is shortly recalled in Aappendix A and gives a Mach stem height in terms of hw and g=w (g is the trailing-edge height). A simple boundary layer model accounting for aerodynamic heating is presented in Appendix
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B. This model provides the displacement thickness of the boundary layer. In Appendix C and Appendix D, deriviation of Dhw and Dðg=wÞ are provided. The displacement wedge angle and trailing-edge height are added to the geometrical wedge angle (hw ) and trailing-edge height (g) to obtain the ðeqÞ equivalent ones (hðeqÞ ). The equivalent parameters hðeqÞ w , g w , ðeqÞ g are used in the inviscid model of Bai and Wu2 to obtain the viscous model. In Section 2.2 we give the magnitudes of Dhw and Dg for typical inflow Mach numbers and Reynolds numbers and study the sensitivity of the variation of Dhw , g on the Mach stem height, to show the importance of viscous effect. We then use the theoretical model to study the influence of Reynolds number on the Mach stem height. A comparison to numerical results by computational fluid dynamics is also provided to assess the applicability of the viscous model.
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2. Mach reflection model with boundary layer correction
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In this section, we first briefly introduce the viscous Mach stem height model, which is based on the inviscid Mach reflection model of Bai and Wu2 and a simplified boundary layer model. The Mach reflection model uses the wedge angle and wedge length as input conditions. A simplified model is built to modify the wedge angle and wedge length in response to the displacement thickness of the boundary layer. The viscous Mach reflection model thus built can then be applied in the same way as the corresponding inviscid model and will be used to study the influence of viscosity in Section 2.2.
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2.1. Summary of viscous Mach reflection model
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The inviscid model gives a Mach stem height HT that varies as HT g ð1Þ ¼ f c; Ma0 ; hw ; w w according to Hornung and Robinson17. It means that the relative Mach stem height is a unique function of the specific heat ratio c, Mach numberMa0 , wedge angle hw and the relative height of the wedge rear corner g=w. The inviscid model of Bai and Wu2 to determine HT is recalled in Appendix A. The essential idea to have a simple viscous Mach reflection model is to assume that, if hw and g=w are replaced by their viscous equivalent ones hðeqÞ and ðg=wÞðeqÞ accounting for the disw placement effect, the Mach stem height still satisfies Eq. (1) so that the inviscid model presented in Appendix A can be directly applied to viscous case, and Eq. (1) becomes g ðeqÞ HT ¼ f c; Ma0 ; hðeqÞ ð2Þ ; w w w where the equivalent wedge angle hðeqÞ is defined as w hðeqÞ ¼ hw þ Dhw w
ð3Þ
and the equivalent relative trailing-edge height ðg=wÞðeqÞ is defined as g ðeqÞ g g ¼ þD ð4Þ w w w Here, Dhw is the displacement wedge angle and Dðg=wÞ is the displacement relative trailing-edge height (see Fig. 2).
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Fig. 2 Mach reflection with viscous boundary layer developed on the wedge lower surface. The boundary layer induces a displacement wedge thickness Dhw and a displacement height Dg for the rear corner R.
Using the displacement thickness d1 ðlÞ given by a simplified boundary layer model as presented in Appendix B, we can estimate Dhw by 8 1:72 ðlaminarÞ < pffiffiffiffiffiffiffi ðcÞ Rew Dhw ð5Þ 0:046 :p ffiffiffiffiffiffiffi ðturbulentÞ 5 ðcÞ Rew
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See Appendix C for details. In Eq. (5), ReðcÞ w is the Reynolds number based on the length w and on the reference temperature, see Appendix B for definition. Due to the curvilinear nature of the boundary layer, the incident shock wave should be also curvilinear. The curvature of the incident shock, not considered here, should have two kinds of influence. The first is the nonuniformity of the flow behind the incident shock wave. The second is the position and local shock angle of the incident shock wave at the reflecting point or triple point (in case of Mach reflection). As a fast model considered in this paper, we simply assume the incident shock wave to be still a straight line, caused by the deflection angle determined by Eq. (3), with Dhw given by Eq. (5). In Appendix D, we give details to derive the expressions for Dðg=wÞ 8 1:72 ðlaminarÞ g < G pffiffiffiffiffiffiffi ðcÞ Rew ð6Þ D 0:046 :Gp ffiffiffiffiffiffiffi w ðturbulentÞ 5 ðcÞ
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where
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Rew
g 1 G ¼ tanhw þ 2coshw w coshw The physical rational range of G is worth discussion. To check the sign of G, we have G wg ; hw ¼ 0 for hw ¼ hðcrÞ w , with the critical wedge angle hðcrÞ satisfying w
g 1 tanhðcrÞ 2coshðcrÞ w þ w ¼ 0 w coshðcrÞ w g For hw < hðcrÞ w , G w ; hw < 0, so that Dðg=wÞ < 0. For g hw > hðcrÞ w , G w ; hw > 0, so that Dðg=wÞ > 0. It appears that g=w would be reduced by boundary layer displacement if hw is small and increased if hw is large. For instance, if we set g=w ¼ 0:4, then hðcrÞ w ¼ 142 . However, hw can never be so large, so that practically we have G < 0. Fig. 3 shows that G < 0 is correct for all reasonable hw . When Ma0 ! 1, obli-
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Fig. 3
Value of Gðg=w; hw Þ for various g=w and hw .
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que shock detachment criterion gives the largest hw (45.4°), which restricts the reasonable maximum value of G. This means that the displacement effect of boundary layer reduces g=w.
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2.2. Influence of boundary layer on Mach stem height
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For purpose of clarifying the importance of boundary layer, the magnitudes of the displacement wedge Dhw angle and displacement relative trailing-edge height Dðg=wÞ for a typical range of Reynolds numbers are computed and the sensitivity of the Mach stem height to the wedge angle and trailingedge height is studied. The viscous Mach reflection model is then used to study the viscous effect (Reynolds number effect) on Mach stem height. These studies are performed using the theoretical model and are presented in Sections 2.2.1–2.2.3. The influence of the boundary layer is further studied using CFD, which is also used to verify the accuracy of the theoretical models. The CFD method is presented in Section 3.1. A comparison of inviscid and viscous CFD solutions is given in Section 3.2. The comparison between theory and CFD is displayed in Section 3.3.
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2.2.1. Magnitudes of displacement wedge parameters Fig. 4 displays the displacement wedge angle Dhw computed by Eq. (5) and the relative displacement trailing-edge height Dðg=wÞ computed by Eq. (7). The Reynolds number Rew ¼ Vm00w varies from 105 to 107. The parameters hw ¼ 22 , g=w ¼ 0:4, Ma0 =2.84 and 4.96 are taken from Hornung and Robinson.17 Consider the boundary layer to be laminar. For the effect of Reynolds number, the absolute value of Dhw and Dðg=wÞ increase when Rew decreases or Ma1 increases, due to thickening of the boundary layer by aerodynamic heating at large Mach number. For instance, when Ma0 = 2.84 is fixed and Rew vary from 107 to 105, Dðg=wÞ changes from 0.0018 to 0.0183 and Dhw changes from 0.1 deg to 0.4°. For turbulent boundary layer, the absolute value of Dhw and Dðg=wÞ is smaller than that for laminar flow when Reynolds number is less than 107. For instance, the quantity Dhw 0.12° and Dðg=wÞ 0.0024 for Rew = 107, and increases to Dhw 0.29° and Dðg=wÞ 0.0055 for Rew = 105.
Fig. 4 Influence of Reynolds number on displacement wedge parameters Dhw and Dðg=wÞ for hw ¼ 22 andg=w ¼ 0:4 at Mach numbers Ma1 = 2.84, 4.96.
Though the amount of Dhw appears to be small, its influence on the Mach stem height will be high, as is clear below.
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2.2.2. Mach stem height for various wedge parameters
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Fig. 5 shows the dependence of the Mach stem height on each factor (hw , g=w), computed by the inviscid Mach refection model of Bai and Wu2, as summarized in Appendix A. To see the influence of hw , we set Ma0 = 2.87, 3.49, 3.98, 4.96 and g=w = 0.4. It is seen that the relative Mach stem height HT =w increases monotonically with hw . When hw changes 1°, HT =w may change by 25% to more than 100%, especially when hw is small. The displacement wedge angle Dhw may be in the order of 1°, as seen from Fig. 4. Hence, it is expected that the viscous boundary layer on the wedge lower surface may have a significant influence on the Mach stem height, especially when hw is small and the Mach number is large. For the influence of the relative trailing-edge height g=w, we set Ma0 = 2.87, 3.49, 3.98, 4.96 and hw ¼ 22 . It is seen that HT =w increases almost linearly with decreasing g=w, for the condition considered here. Thus, the Mach stem height is less sensitive to g=w compared to hw . Thus the influence of the boundary layer on the Mach stem height is majorly due to the increase of the equivalent wedge angle.
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Fig. 5
Mach stem height.
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2.2.3. Effect of Reynolds number
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Now we study the influence of viscous effect, in terms of the Reynolds number, on the Mach stem height using the present viscous Mach reflection model (Section 2.1). The inviscid model of Bai and Wu2 as summarized in Section 2.1 is applied by replacing hw and g=w with the equivalent hðeqÞ and ðg=wÞðeqÞ , w ðeqÞ see Eqs. (3) and (5) for hw , Eqs. (4) and (6) for ðg=wÞðeqÞ . The boundary layer is either assumed fully laminar or fully turbulent, though in real applications the boundary layer may be mixed laminar-turbulent. Fig. 5(b) gives the Mach stem height for Reynolds number Rew ranging from 105 to 107, with hw ¼ 22 , Ma0 =4.96, g=w=0.4. Compared to inviscid mode, the increase of Mach stem height due to viscous effect is large, especially for small Reynolds number and for turbulent boundary layer. For instance, at Rew = 106, the viscous model gives a Mach stem height HT =w0.034 for turbulent boundary layer and HT =w0.033 for laminar boundary layer, compared to HT =w0.029 for inviscid flow. For small Reynolds number, the turbulent model predicts a Mach stem height smaller than
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the laminar one, noting that for such small Reynolds number the boundary layer should be laminar. For Rew > 1:04 107 the turbulent model gives a larger Mach stem height, conversely. When Reynolds number tends to infinity, boundary layer thickness tends to zero according to Eq.(B1), and there is no difference between inviscid, laminar and turbulent model’s prediction.
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3. Results analysis
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3.1. Method and boundary conditions for numerical computation
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For numerical simulation, we use the experimental conditions of Ivanov et al.18 who measured the transition for symmetric reflection in a low-noise wind tunnel. Specifically, we use Ma0 = 4, g=w=0.43 and Rew = 3.83 105 (the Reynolds number based on unit length is 1.278 107). The temperature is T0 = 65 K and the pressure is 1641 Pa. Note that Ivanov et al.18 did not provide these conditions for temperature and pressure and we find these conditions from their previous work. Two wedge angles, hw ¼ 23:21 ; 24:60 , are used for inviscid and viscous computations. For numerical simulations of inviscid flow, the compressible Euler equations in gas dynamics are solved using the second order Roe scheme.19 For viscous flow, the compressible Navier-Stokes equations are solved, with the inviscid part also solved using the second order Roe scheme and with the viscous part solved using the second order central difference scheme. For turbulence modelling in viscous flow computation, the SST two equation model20 is used. All these methods exist in the commercial code Fluent and it is this code that is used for CFD simulation here. The grid used contains 1600 points along the horizontal direction and 700 points along the vertical direction. This grid is much denser than the grid used before by other authors. For instance, Gao and Wu7 used a grid with 300 200 points and Wu et al.21 used grid with 1540 400 points. The grid is refined near the triple point structure and near the wedge lower surface to capture the boundary layer. As displayed in Fig. 2 for boundary conditions, Q1Q2 is a supersonic inlet, AQ1 is a symmetry, Q2Q3 is also a symmetry (reflecting surface), no-slip and adiabatic wall condition is prescribed along AR and RQ₅, Q3Q₄ is a supersonic outlet, and Q₄Q₅ is a nonreflecting boundary. Note that RQ₅ is the backside of the wedge similar to the geometry of Ivanov et al.18 who used a triangular wedge and Q₅ is the upper rear corner of this wedge.
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3.2. Comparison between inviscid and viscous numerical results
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Fig. 6 displays the Mach contours obtained by both inviscid and viscous simulations (the same hw is used). Differences of shock angles are observed for inviscid and viscous flows. The shock angle of the incident shock wave is larger in viscous flow than that in inviscid flow. The Mach stem height in viscous flow is higher than that in inviscid flow, as predicted theoretically above. Shock structures given by current viscous theory are also displayed, and match well with viscous numerical results. Note that there is also a boundary layer along RQ₅ for viscous flow. To see the influence of this boundary layer, we also
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Fig. 6 Mach contours by inviscid and viscous flow computations for Ma0 = 4.
Fig. 7 Mach contours by viscous flow computations with RQ₅ treated as wall and non-reflecting boundary for Ma0 = 4 and hw ¼ 23:21 .
Fig. 8 Comparison of Mach contours between the viscous CFD result (with hw ¼ 23:21 ) and modified inviscid CFD result (hw ¼ 23:21 þ Dhw , g=w ¼ 0:43 þ Dðg=wÞ, where Dhw ¼ 0:23 and Dðg=wÞ ¼ 0:0022) for Ma0 = 4.
Fig. 9 Comparison between theory and CFD simulation for Mach stem height at Ma0 = 4, g=w ¼ 0:43 and Rew = 3.83 105.
computed the flow where RQ₅ is replaced by a non-reflecting wall. Fig. 7 displays a comparison of Mach contours with RQ₅ treated as a wall and a non-reflecting boundary, for vis cous flow with hw ¼ 23:21 . It is seen that the Mach stem height is almost the same for both treatments of RQ₅, at least for the conditions considered. It is interesting to see whether the viscous numerical result can be reproduced through inviscid one with correction of the wedge parameters. In Fig. 8, we compare, for Ma0 = 4, hw ¼ 23:21 , viscous CFD result with an inviscid one where hw is augmented by Dhw through Eq.(5) (Dhw ¼ 0:23 in this case) and g/w is changed by Dðg=wÞ through Eq. (6) (Dðg=wÞ ¼ 0:0022 in this case). The close agreement indicates
Table 1 Comparison between theory and numerical simulation for Ma0 = 4, Rew =3.83 105 and g=w ¼ 0:43. The theoretical results are obtained by the model of Bai and Wu (2017)2 for inviscid flow and the present viscous model for viscous flow. Condition
CFD
Theory
Inviscid
Viscous
Inviscid
Viscous
hw ð Þ
p2 =p1
bð Þ
HT =w
bð Þ
HT =w
bð Þ
HT =w
bð Þ
HT =w
23.21 24.60
6.36 6.89
36.23 37.86
0.0878 0.1582
36.77 38.40
0.0989 0.1696
36.24 37.95
0.0870 0.1540
36.52 38.24
0.1021 0.1725
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Fig. 10 Influence of viscous effect on Mach stem height at Ma0 = 4.96, g=w ¼ 0:4 and Rew = 1.36 10⁶.
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that a simple boundary model correcting the inviscid model is suitable for the present study.
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3.3. Comparison between theory and numerical results
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Table 1 displays the comparison between theory and numerical results for the shock angle and relative Mach stem height. For a given hw and Ma0 , the shock angle b of inviscid flow is computed by the shock angle relation: tanhw ¼
2 tanb
Ma20 sin2 b 1 Ma20 ðc þ cos2bÞ þ
2
For viscous flow, the Eq. (5) is used to find the displacement wedge angle Dhw and then hw is replaced by hw þ Dhw in the above relation to obtain the shock angle in viscous flow. As shown in Table 1, the shock angles obtained using such simplified models match very well with the CFD results. The displacement effect of the boundary layer enlarges the shock angle by an amount close to 0:5 . Good comparison is obtained for the relative Mach stem height HT =w, according to Table 1 and Fig. 9. Fig. 9(a) displays the Mach stem height for hw and shows difference of HT =w at any given hw . Both theory and CFD shows that the Mach stem height is increased by
Fig. 11 model.
Further comparison and validation for current viscous
viscous boundary layer developed on the lower wedge surface. Fig. 9(b), which displays the Mach stem height for the shock angle, shows no difference between inviscid and viscous models, since the displacement effect of the boundary layer is automatically included in the shock angle. The conclusion is similar for other conditions, as displayed in Fig. 10 for Ma0 = 4.96, Rew =1.36 106 g=w ¼ 0:4 and Fig. 11(b) for various Mach and Reynolds number. The condition Ma0 = 3.98 and g=w ¼ 0:4 has been used by various authors for comparison. The comparison of the viscous Mach reflection model with some previous works is displayed in Fig. 11, where we also display the experimental data by Mouton and Hornung,22 the numerical data from inviscid computational fluid dynamics by Mouton and Hornung.6 The predictions by inviscid models of Gao and Wu,7 Mouton and Hornung6 and Bai and Wu2 are also displayed.
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Z. CHEN et al. Table 2
Comparison between theory and previous experimental/numerical results for various conditions.
Condition
Reference result
Theory
hw (°)
g=w
p2 =p1
bð Þ
HT =w
bð Þ
HT =w
2 10 2 106 2 106
23.0 25.0 27.0
0.34 0.34 0.34
8.53 9.68 10.91
33.5 35.8 37.1
0.0493 0.1151 0.2137
33.46 35.91 38.4
0.0649 0.1439 0.2579
3.98 3.98 3.98
1:33 106 1:33 106 1:33 106
21.4 22.3 23.0
0.4 0.4 0.4
5.66 5.98 6.23
34.1 35.12 36.2
0.0271 0.0467 0.0993
34.0 35.2 36.15
0.0269 0.0554 0.0905
3.98 3.98 3.98
1:33 106 1:33 106 1:33 106
22.5 23.0 23.5
0.4 0.4 0.4
6.05 6.23 6.42
35.44 36.03 36.66
0.0642 0.0869 0.1077
35.45 36.04 36.67
0.0658 0.0844 0.1066
Ma0
Rew
Schmisseur (2011)
4.96 4.96 4.96
Mouton and Hornung (2008)22
Mouton and Hornung (2007)6
14
6
408
Comparison between theory and previous experimental/ numerical results for various conditions is shown in Table 2.
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4. Conclusions
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Using a very simple boundary layer model to compute the displacement thickness, we have obtained approximate expressions for the displacement wedge angle and displacement trailing edge height. These displacement parameters are used to correct the geometrical wedge angle and trailing-edge height in the inviscid Mach reflection model of Bai and Wu2. The viscous model thus obtained compares well with the present numerical computation using a very fine grid. The present study shows that the displacement effect of the boundary layer developed over the lower wedge surface has an important effect on the Mach stem height, even though the displacement thickness is very small. It increases the Mach stem height through increasing the effective wedge angle. A slight change of this effective parameter may change, in some conditions, significantly the Mach stem height. Acknowledgement
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This work was supported by the Natural National Science Foundation of China (No. 11802157), Special Foundation of Chinese Postdoctoral Science (No. 2019T120082), and Chinese Post-doc Science Foundation (No. 2018M640119).
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Appendix A. Summary of the inviscid model
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431 432 433 434 435 436 437 438 439 440 441 442 443 444
As shown in Fig. 1, the three shock waves and slipline separate the flow into four regions i = 0,1,2,3. In the following we use subscript i to denote solution in region i and superscript T to denote the solution in the vicinity of the triple point. Using the three shock theory of von Neumann gives the shock angles b1 (incident shock wave), bTr (reflected shock wave) and bTm (Mach stem), the Mach numbers Ma1 , MaT2 and MaT3 , the pressures p1 , pT2 and pT3 , and the initial angle of the slipline dTs . These parameters are independent of the Mach stem height. In Eq. (1), the Mach stem height HT should be first guessed and then evaluated through an iterative process. The simplest initial guess is to put HT ¼ g=2.
We choose a coordinate system such that the axis x is along the reflecting surface and the axis y passes the triple point, i.e., the triple point is at (0, HT ), as shown in Fig. 1. For such a coordinate system, the position of the trailing edge R is given by xR ¼ wcoshw ðHA HT Þcotb1 yR ¼ g where HA is the inlet height satisfying HA ¼ g þ wsinhw . The first characteristics of the wedge expansion fan intersects the reflected shock wave at point F. Assuming the shock wave TF to be straight, the position (xF , yF ) can be determined by yF yT ¼ ðxF xT Þtan bTr hw ðA1Þ yF yR ¼ ðxF xR Þtanðl1 þ hw Þ Here l1 ¼ arcsinð1=Ma1 Þ is the Mach angle in region 1. The inviscid model of Bai and Wu2 contains an analytical model for the free part of the slipline, one model for the reflected shock wave and one model for the interacting part of the slipline. The slipline model requires the inlet condition of the quasione-dimensional duct to be specified. The simplest way is to 1 T N N put Mam ¼ 12 MaT3 þ MaN g ,pm ¼ 2 p3 þ pg , where Mag and pN g are normal shock solutions of the Mach number and pressure at the foot of the Mach stem. The shape (height Hs ) of the free part TB of the slipline is given by Hs HT
þ 2ðcþ1 Mam c1Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # Mas c1 c þ 2 #ðMas Þ1 c1 #ðMam ÞN
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cþ1 N 2c
N¼
pT 2 pm
ðA2Þ
c T c1 # Ma2
Maþ s
Here, is the Mach number just above the slipline and is related to the deflection angle of the slipline ds ¼ arctanðdHs =Hs Þ by the Prandtl-Meyer function qffiffiffiffiffiffi ¼ ds dTs þ v MaT2 . Here vðMaÞ ¼ cþ1 arctan v Maþ s c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c1 ðMa2 1Þ arctan Ma2 1 and c þ 1 #ðMaÞ ¼ 1 þ c1 Ma2 . Eq. (A2) was obtained by pressure bal2 ance between the quasi-one-dimensional duct and secondary expansion waves over the slipline. The interacting part (BE) of the slipline depends on the shape of the reflected shock wave. Now consider the shape of the reflected shock wave. Let hf be the deflection angle of the flow stream in the expansion fan,
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with hf ¼ hw for the leading characteristics RF. When the flow direction deflects downwards, hf is considered positive. For any given hf , the Mach number Mf and the pressure pf in the expansion fan follow from the Prandtl-Meyer solution. The interacting part (segment FK in Fig. 1) of the reflected shock wave has a shock angle br following the expression dbr dhf
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¼
ðbh K5 bM Þbh K4 K3 1þ2bh K4 sinð2br Þ
br ¼ bTr at hf ¼ hw
The abscissa xr on the reflected shock is connected to hf through dxr dhf tan2
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¼
1Ma2f #ðMaf Þ
524 525
Ma2f 1
ðhf þlf Þþ1
tanðbr hf Þþtanðhf þlf Þ
ðA4Þ
ðxr xR Þ
where lf is the Mach angle at Maf . The parameters involved in (A4) are defined by 8
1 > bh ¼ 1 K1 þ 2K2 Maf cos2 br > > > > 1 > > bM ¼ ½1 K1 þ 2K2 cos2 br K2 sinð2br Þ > > > > > sin2ðb hf þht Þ > > K1 ¼ sinrð2br Þ > > > > < 2ðcþ1ÞMaf sin2 ðbr hf þht Þ i2 K2 ¼ h > ðc1ÞMa2f sin2 br þ2 > > > > > Maf > > ffi # Maf K5 ¼ pffiffiffiffiffiffiffiffiffiffi > > Ma2f 1 > > > > > > Ma2 sin2 br 1 > > ht ¼ hf arctan 2cotbr 2 f : Starting from point F, where the shock angle is br ¼ bTr , dyr solve Eqs. (A3), (A4) and dx ¼ tan br hf to find r br ¼ br hf and the position of the reflected shock wave (xr , yr ) for hf varying from hw to 0. At any hf or (xr ,yr ), the quantities Mat and pt (solutions just downstream of the reflected shock wave) are connected to Maf and pf through the oblique shock wave expressions, with the shock angle br . The interacting part of the slipline starts at the intersection point B (called turning point since it was found that the slipline changes slope here) of the slipline and the first characteristics of the transmitted expansion waves. The position (xB , yB ) of B can be determined through Eq. (A3) and yB yF ¼ ðxB xF ÞtanðlF þ ht;F Þ where lF ¼ arcsinð1= MaT2 Þ and ht;F ¼ dTs . For the interacting point of the slipline (BE), the deflection angle of the slipline ds satisfies dds cpf Ma2f ðK6 þ K7 Þ þ Wðds Þ1 dxs cþ1
cMa2s p þ s tands ¼ 0 1 Ma2s Hs where
1c2sin2 br 2Ma2f ffi pffiffiffiffiffiffiffiffiffiffi K6 ¼ 2 Maf 1
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¼ Wðds Þ
Wðds Þ ¼
ðK6 þK7 Þpf Ma2f
pffiffiffiffiffiffiffiffiffiffi ffi 2 Mat 1
ðcþ1Þpt Ma2t
ðA6Þ
K
531
where K¼
532 533
tan ðlt þ ht Þ þ 1 tanðlt þ ht Þ tands 2
Ma2t
1 þ #ðMat Þ ðxs xG Þ Ma2t 1
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The parameters with subscript t have been given in the step for the reflected shock wave. The flow parameters below the slipline are related to Hs ¼ Hs ðxÞ through the quasi-onedimensional isentropic flow relations. Starting from point B, and knowing Mat and pt from the solution of the shock/expansion wave interaction algorithm, the slipline/transmitted expansion wave interacting Eqs. (A5) dds and (A6) are used to find xs , ds ¼ ds ðxs Þ, dx and s Hs ¼ Hs ðxs Þ, for any hf varying from hw to 0. This gives one throat position xd , with ds ðxd Þ ¼ 0. Use the quasi-one-dimensional isentropic relation to find the Mach number Mas in the quasi-one-dimensional duct. This gives another throat position xM , with Mas ðxM Þ ¼ Mas ¼ 1. If xd ¼ xM , then the initial choice of the Mach stem height is correct and is said to meet the sonic throat compatibility condition. If xd –xM , then the choice of the Mach stem height xT is incorrect and should be updated until xd ¼ xM . Bai and Wu2 updated this height using the method of bisection. For given Ma0 , g=w and hw , the height of the Mach stem is unique.
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Appendix B. The boundary layer model
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The displacement thickness of the boundary layer developed over the lower surface of the wedge is used here to account for the viscous effect on Mach reflection. The Mach number may be high enough so that boundary layer is affected by aerodynamic heating. This heating increases the boundary layer thickness in practical application though in many wind tunnel experiments the inflow temperature is very low. For both laminar and turbulent flow, a simple idea to account for aerodynamic heating is to use the reference temperature method. In this method, the formulas for boundary layer parameters are taken from incompressible flow, wherein the thermodynamic and transport properties such as density and viscosity in these formulas are evaluated at some reference temperature T that represents the temperature somewhere inside the boundary layer.15 This method was originally developed by Rubesin and Johnson23 and then advanced by various authors including Eckert,24 Eckert and Tewfik,25 Ott and Anderson,26 Dorrance,27 Herwig28 and Meador and Smart,29 see van Oudheusden30 for a review. In this paper, we assume the boundary layer is purely laminar or purely turbulent. For intermediate Reynolds numbers, the boundary layer may be mixed, i.e., the boundary layer near the leading edge is laminar while after a transition point the boundary layer is turbulent. The case of mixed turbulent boundary layer is not considered in this paper. Here we use l to measure the distance along the lower surface of the edge, starting from the leading edge (A). Various simplified models have been developed for evaluating the displacement thickness d1 ðlÞ of boundary layer (see for instance Schlichting31). Let Rel be the Reynolds number based
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Maf ðcþcos2br Þþ2
cpt Ma2t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ma2t 1 523
ðA3Þ
dxs dhf
9
K7 ¼
K5 bM Þbh K4 K3 2sinð2br Þ ðbh1þ2b h K4 sinð2br Þ
The abscise xs of the slipline is related to hf by
ðA5Þ
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on the inflow conditions in region (0) and on the distance l. For incompressible flow d1 ðlÞ can be evaluated by 8 1:72l ðlaminarÞ < pffiffiffiffiffi Rel d1 ðlÞ 0:046l :p ffiffiffiffiffi ðturbulentÞ 5 Rel
For compressible flow, the reference temperature is used to evaluate the Reynolds number, in such a way that the displacement thickness is evaluated by 8 1:72l ðlaminarÞ > < pffiffiffiffiffiffiffi ðcÞ Rel d1 ðlÞ ðB1Þ 0:046l > ffiffiffiffiffiffiffi ðturbulentÞ :p 5 ðcÞ Rel
ðcÞ
Here Rel ¼ Rel =c is the Reynolds number based on the reference temperature, and c ¼ TT0 llððTT0 ÞÞ is the ChapmanRobinson number. The temperature-dependent viscosity coefficient l is estimated by the Sutherland law 1:5 Ts þ C T lðTÞ ¼ 1:716 105 Pa s TþC C where Ts ¼ 273:15 K and C ¼ 110:4 K, and the reference temperature T is evaluated using the most popular formula by Eckert.24
T 0:5T0 þ 0:5Tw þ 0:22ðTr T0 Þ
ðB2Þ
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Here Tw is the wall temperature and Tr ¼ T0 ð1þ is the recovery temperature, R is the recovery factor, with R 0:85 for laminar boundary layer and R 0:89 for turbulent boundary layer. In this paper we assume an adiabatic wall so that Tw ¼ Tr .
615
Appendix C. Derivation of hw
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616 617 619 620 621 622 624 625 626 627
629 630
631
632 633
635
R c1 Ma20 Þ 2
The local effective angle ðeqÞ hloc ðlÞ
ðeqÞ hloc ðlÞ
at l is here defined as
dd1 ðlÞ ¼ hw þ dl
and hwðeqÞ is evaluated by average hðeqÞ ¼ w1 w R w dd1 ðlÞ ðeqÞ 1 hw ¼ hw þ w 0 dl dl and Z 1 w dd1 ðlÞ Dhw ¼ dl w 0 dl
Rw 0
dd1 ðlÞ dl. dl
This yields
ðC1Þ
Introducing the Eq. (B1) in Appendix B for d1 ðlÞ into (B1) gives 8 1:72 ðlaminarÞ < pffiffiffiffiffiffiffi ðcÞ Rew Dhw ðC2Þ 0:046 :p ffiffiffiffiffiffiffi ðturbulentÞ 5 ðcÞ Rew
Here ReðcÞ w is the Reynolds number
ðcÞ Rel
at l ¼ w.
Appendix D. Derivation of ðg=wÞ Now we derive Dðg=wÞ by using the obvious relations 8 ðeqÞ ðeqÞ > < L ¼ w coshw ¼ wcoshw HA gðeqÞ ¼ wðeqÞ sinðhw þ Dhw Þ > : HA g ¼ wsinhw
ðD1Þ
Assuming Dhw to be small relative to hw , the first relation of Eq. (D1) gives coshw w w wDhw tanhw cosðhw þ Dhw Þ
wðeqÞ
ðD2Þ
and the second relation of Eq. (D1) gives ðeqÞ
HA g wðeqÞ
ðeqÞ
HA g wðeqÞ
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sinhw þ Dhw coshw
Using the third relation of Eq. (D1) we sinhw ¼ ðHA gÞ=w so that the above relation yields
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644
get
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HA g þ Dhw coshw w
649
The above relation can be combined with (4) in Section 2.1 to give g HA HA Dhw coshw þ ðeqÞ D w w w
650
which, if Eq.(3) in Section 2.1 is used, becomes g HA Dhw coshw þ Dhw tanhw D w w
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ðD3Þ
Putting HA ¼ g þ wsinhw into Eq. (D3) and noting that coshw þ sinhw tanhw ¼ ðcoshw Þ1 2coshw we get g g 1 tanhw þ D 2coshw Dhw w w coshw Replacing Dhw by Eq. (C2) in the above expression finally gives Eq. (6) in Section 2.1.
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References
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1. Ben-Dor G. Shock wave reflection phenomena. Springer; 2007. 2. Bai CY, Wu ZN. Size and shape of shock waves and slipline for Mach reflection in steady flow. J Fluid Mech 2017;818:116–40. 3. Mach E. Uber den verlauf von Funkenwellen in der Ebene und im Raume. Sitzungsber Akad Wiss Wien 1978;1878:819–38. 4. Azevedo DJ, Liu CS. Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA J 1993;31 (1):83–90. 5. Li H, Ben-Dor G. Application of the principle of minimum entropy production to shock wave reflections. I. Steady flows. J Appl Phys 1996;80(4):2027–37. 6. Mouton CA, Hornung HG. Mach stem height and growth rate predictions. AIAA J 2007;45:1977–87. 7. Gao B, Wu ZN. A study of the flow structure for Mach reflection in steady supersonic flow. J Fluid Mech 2010;656:29–50. 8. Von Neumann J. Refraction, intersection and reflection of shock waves. Washington, D.C.: Navy Dept. Bureau of Ordinance; 1945. 9. Courant R, Friedrichs KO. Supersonic flow and shock waves. Wiley-Interscience; 1948. 10. Liepmann HW, Roshko A. Elements of gasdynamics. New York: John Wiley and Sons; 1957. 11. Emanuel G. Gasdynamics: theory and applications. Reston: AIAA; 1986. 12. Ben-Dor G, Takayama K. The phenomena of shock wave reflection-a review of unsolved problems and future research needs. Shock Waves 1992;2:211–23. 13. Li H, Ben-Dor G. A parametric study of Mach reflection in steady flows. J Fluid Mech 1997;341:101–25. 14. Schmisseur JD, Gaitonde DV. Numerical simulation of Mach reflection in steady flows. Shock Waves 2011;21:499–509. 15. Anderson JD. Hypersonic and high temperature gas dynamics. 2nd ed. New York: Mcgraw-Hill Book Company; 2006.
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16. Babinsky H, Harvey JK. Shock wave boundary layer interaction. New York: Cambridge University Press; 1989. 17. Hornung HG, Robinson ML. Transition from regular to Mach reflection of shock waves. Part 2. The steady-flow criterion. J Fluid Mech 1982;123:155–64. 18. Ivanov MS, Kudryavtsev AN, Nikiforov SB, Khotyanovsky DV, Pavlov AA. Experiments on shock wave reflection transition and hysteresis in low-noise wind tunnel. Phys Fluids 2003;15 (6):1807–10. 19. Roe PL. Characteristic based schemes for the Euler equations. Annu Rev Fluid Mech 1986;18:337–65. 20. Menter FR. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 1994;32(8):1598–605. 21. Li SG, Gao B, Wu ZN. Time history of regular to Mach reflection transition in steady supersonic flow. J Fluid Mech 2011;682:160–84. 22. Mouton CA, Hornung HG. Experiments on the mechanism of inducing transition between regular and Mach reflection Mach stem height and growth rate predictions. Phys Fluids 2008;20 (8):126103. 23. Rubesin MW, Johnson HA. A critical review of skin-friction and heat-transfer solutions of the laminar boundary layer of a flat plate. Trans Am Soc Mech Eng 1949;71:383–8.
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24. Eckert ERG. Engineering relations for heat transfer and friction in high-velocity laminar and turbulent boundary-layer flow over surfaces with constant pressure and temperature. Trans. Am. Soc Mech Eng 1956;78:1273–83. 25. Eckert ERG, Tewfik OE. Use of reference enthalpy in specifying the laminar heat-transfer distribution around blunt bodies in dissociated air. J Aerospace Sci 1960;27(6):464–6. 26. Ott JD, Anderson JD. Effects of nonequilibrium chemistry on the reference temperature method and Reynolds analogy. J Thermophys Heat Transfer 1994;8(2):381–4. 27. Dorrance WH. Viscous hypersonic flow. New York: McGraw-Hill; 1962. p. 134–40. 28. Herwig H. An asymptotic approach to compressible boundarylayer flow. Int J Heat Mass Transfer 1987;30:59–68. 29. Meador WE, Smart MK. Reference enthalpy method developed from solutions of the boundary layer equations. AIAA J 2005;43 (1):135–9. 30. van Oudheusden BW. Some classic thermal boundary layer concepts reconsidered. In: Meier GEA, Sreenivasan KR, editors. IUTAM symposium on one hundred years of boundary layer. Netherlands: Springer; 2006. p. 425–34. 31. Schlichting H. Boundary-layer theory. 7th ed. New York: McGraw Hill; 1979.
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No. of Pages 11
1
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Mach reflection in steady supersonic flow considering wedge boundary-layer correction
5
Zijun CHEN a, Chenyuan BAI a,b, Ziniu WU a,*
3
6 7
8
a b
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Institute of Aeroengine, Tsinghua University, Beijing 100084, China
Received 14 December 2018; revised 2 January 2019; accepted 26 February 2019
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KEYWORDS
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Boundary-layer correction; Mach reflection; Shock waves; Supersonic flow; Viscous effect
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Abstract Mach reflection in steady supersonic flow is an important phenomenon having received extensive studies, among which simplified theoretical models to predict the size of Mach stem and other flow structure are of particular interest. Past efforts for such models were based on inviscid assumption while in real cases the flow is viscous. Here in this paper we consider the influence of wedge boundary layer on the Mach stem height. This is done by including a simplified boundary layer model into a recently published inviscid model. In this viscous model, the wedge angle and the trailing edge height, which control the Mach stem height, are replaced by their equivalent ones accounting for the displacement effect of the wedge boundary layer, with the boundary layer assumed to be laminar or fully turbulent. This viscous model is shown to compare well with numerical results by computational fluid dynamics and gives a Mach stem height as function of the Reynolds number and Mach number. It is shown that due to the viscous effect, the Mach stem height is increased, through increasing the effective wedge angle. Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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Shock reflection in steady supersonic flow is a well-studied phenomenon.1 As shown in Fig. 1,2 a wedge of length w and angle hw induces an incident shock wave (i), which reflects from the reflecting surface. When the wedge angle is large enough or
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* Corresponding author. E-mail address: [email protected] (Z. WU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
the inflow Mach number Ma0 is small enough, the reflected shock wave detaches and a special reflection configuration, called irregular reflection or Mach reflection (since this was first observed by Mach3), occurs. Mach reflection has a three-shock structure. A triple shock point (T) connects these three shock waves, including the incident shock wave (i), the reflected shock wave (r) and a strong shock wave called Mach stem (m). A slipline (s), generated from the triple point, separates the flows downstream of the reflected shock wave and Mach stem and across this slipline the pressure is balanced. The slipline and the reflecting surface enclose a flow region that can be regarded as quasi-onedimensional.4–7 The transmitted expansion waves from the trailing-edge (R) intersect this slipline and reduces the pres-
https://doi.org/10.1016/j.cja.2019.09.007 1000-9361 Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: CHEN Z et al. Mach reflection in steady supersonic flow considering wedge boundary-layer correction, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.09.007
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Fig. 1 Mach reflection configuration following Bai and Wu.2 The reflection of the incident shock wave (i) produces a reflected shock wave (r) composed of a free segment (TF), an interacting segment (FK) and another straight segment (r’), a strong shock wave called Mach stem (m), and a slipline (s) (composed of a free segment TB and an interactive segment BE). Secondary Mach waves are generated over the slipline. A sonic throat exists in the quasi-one-dimensional flow region below the slipline.
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sure, so that the quasi-one-dimensional region or duct has a sonic throat. The three shock waves and the slipline separate the flow into four regions, the solutions of which in the vicinity of the triple point can be found using the three-shock theory of von Neumann.8 The mechanism by which the size of the Mach stem is determined has been a long-standing problem. This problem was initially raised by Courant and Friedrichs9 (see also Liepmann and Roshko10) and about forty years later it was still regarded as unsolved by Emanuel,11 see also Ben-Dor and Takayama.12 Azevedo and Liu4 proposed for the first time a physical model for predicting the Mach stem height. In this model, the sonic throat is assumed to occur where the leading Mach wave of the expansion fan intersects the slip stream and the flow below the slipline is treated using the isentropic quasi-onedimensional ideal gas flow theory. Li and Ben-Dor13 allowed the sonic throat to occur further downstream and assumed that the Mach stem (m), the reflected shock wave (r) and the slipline (s) are slightly curved to be modelled by second order polynomials. Mouton and Hornung6 constructed a model for the growth rate of the Mach stem height during the transition and obtained the Mach stem height as the transition is finished. They assumed that the Mach stem, the reflected shock wave, the slipline and the Mach waves to be straight lines. Gao and Wu7 observed from numerical simulation that secondary Mach waves are generated over slipline to balance the pressure decrease below the slipline and built a characteristic theory to account for the influence of these secondary waves. Bai and Wu2 further derived analytical expressions for the shape of slipline and reflected shock wave and found that the slipline changes slope at the turning point (B) where the leading Mach wave of the expansion fan intersects the slip stream. They then believed that a turning point wave, in the form of a weak shock wave, is generated from this turning point. Recently, Schmisseur and Gaitonde14 performed a numerical study of the Mach stem height taking into account the effect of viscosity. Mach stem heights from inviscid and viscous flow calculations using Computational Fluid Dynamics (CFD) are compared and it was found that viscosity increases the Mach stem height. They pointed out that the increase of
the Mach stem height is due to the increased displacement effect of the wedge boundary-layer and suggested that further work is essential to confirm this hypothesis. In both experimental and numerical studies for the Mach stem height in viscous flow conditions, one often uses the shock angle as input condition, making the influence of viscosity unclear. The boundary layer developed over the lower surface of the wedge not only increases the effective wedge angle and but also reduces the effective height of the wedge trailing edge. Measuring the shock angle automatically includes the influence of the boundary layer in increasing the shock angle, but the effect of the reduced effective trailing-edge height is not considered. Though the displacement thickness of boundary layer is small so that the increase of the effective wedge angle and the decrease of the effective trailing-edge height may be small, its influence on Mach stem may be large since the dependence of the Mach stem height on the geometrical parameters is very sensitive, especially when the Mach stem height is small.13 The purpose of this paper is therefore to built a viscous Mach reflection model for Mach stem height, to account for the influence of the boundary layer developed over the lower surface of the wedge. This influence is expected to be important when the Reynolds number is low (since the displacement thickness of the boundary layer increases for decreasing Reynolds number) or when the Mach number is large (since aerodynamic heating for large Mach number increases the boundary layer thickness15). In the present paper, the reflecting surface is considered as an inviscid one (as in the case of the symmetrical plane of Mach reflection by intersection of two shock waves from opposite sides). For viscous reflecting surface as in the case of a solid wall, the problem is complicated by shock wave/boundary layer interaction.16 In Section 2.1, the viscous model based on the boundary correction to the inviscid model of Bai and Wu2 is introduced. This model gives the displacement wedge angle (Dhw ) and equivalent relative trailing-edge height (Dðg=wÞ) for modification. For details, the inviscid model is shortly recalled in Aappendix A and gives a Mach stem height in terms of hw and g=w (g is the trailing-edge height). A simple boundary layer model accounting for aerodynamic heating is presented in Appendix
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B. This model provides the displacement thickness of the boundary layer. In Appendix C and Appendix D, deriviation of Dhw and Dðg=wÞ are provided. The displacement wedge angle and trailing-edge height are added to the geometrical wedge angle (hw ) and trailing-edge height (g) to obtain the ðeqÞ equivalent ones (hðeqÞ ). The equivalent parameters hðeqÞ w , g w , ðeqÞ g are used in the inviscid model of Bai and Wu2 to obtain the viscous model. In Section 2.2 we give the magnitudes of Dhw and Dg for typical inflow Mach numbers and Reynolds numbers and study the sensitivity of the variation of Dhw , g on the Mach stem height, to show the importance of viscous effect. We then use the theoretical model to study the influence of Reynolds number on the Mach stem height. A comparison to numerical results by computational fluid dynamics is also provided to assess the applicability of the viscous model.
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2. Mach reflection model with boundary layer correction
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In this section, we first briefly introduce the viscous Mach stem height model, which is based on the inviscid Mach reflection model of Bai and Wu2 and a simplified boundary layer model. The Mach reflection model uses the wedge angle and wedge length as input conditions. A simplified model is built to modify the wedge angle and wedge length in response to the displacement thickness of the boundary layer. The viscous Mach reflection model thus built can then be applied in the same way as the corresponding inviscid model and will be used to study the influence of viscosity in Section 2.2.
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2.1. Summary of viscous Mach reflection model
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The inviscid model gives a Mach stem height HT that varies as HT g ð1Þ ¼ f c; Ma0 ; hw ; w w according to Hornung and Robinson17. It means that the relative Mach stem height is a unique function of the specific heat ratio c, Mach numberMa0 , wedge angle hw and the relative height of the wedge rear corner g=w. The inviscid model of Bai and Wu2 to determine HT is recalled in Appendix A. The essential idea to have a simple viscous Mach reflection model is to assume that, if hw and g=w are replaced by their viscous equivalent ones hðeqÞ and ðg=wÞðeqÞ accounting for the disw placement effect, the Mach stem height still satisfies Eq. (1) so that the inviscid model presented in Appendix A can be directly applied to viscous case, and Eq. (1) becomes g ðeqÞ HT ¼ f c; Ma0 ; hðeqÞ ð2Þ ; w w w where the equivalent wedge angle hðeqÞ is defined as w hðeqÞ ¼ hw þ Dhw w
ð3Þ
and the equivalent relative trailing-edge height ðg=wÞðeqÞ is defined as g ðeqÞ g g ¼ þD ð4Þ w w w Here, Dhw is the displacement wedge angle and Dðg=wÞ is the displacement relative trailing-edge height (see Fig. 2).
3
Fig. 2 Mach reflection with viscous boundary layer developed on the wedge lower surface. The boundary layer induces a displacement wedge thickness Dhw and a displacement height Dg for the rear corner R.
Using the displacement thickness d1 ðlÞ given by a simplified boundary layer model as presented in Appendix B, we can estimate Dhw by 8 1:72 ðlaminarÞ < pffiffiffiffiffiffiffi ðcÞ Rew Dhw ð5Þ 0:046 :p ffiffiffiffiffiffiffi ðturbulentÞ 5 ðcÞ Rew
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182
See Appendix C for details. In Eq. (5), ReðcÞ w is the Reynolds number based on the length w and on the reference temperature, see Appendix B for definition. Due to the curvilinear nature of the boundary layer, the incident shock wave should be also curvilinear. The curvature of the incident shock, not considered here, should have two kinds of influence. The first is the nonuniformity of the flow behind the incident shock wave. The second is the position and local shock angle of the incident shock wave at the reflecting point or triple point (in case of Mach reflection). As a fast model considered in this paper, we simply assume the incident shock wave to be still a straight line, caused by the deflection angle determined by Eq. (3), with Dhw given by Eq. (5). In Appendix D, we give details to derive the expressions for Dðg=wÞ 8 1:72 ðlaminarÞ g < G pffiffiffiffiffiffiffi ðcÞ Rew ð6Þ D 0:046 :Gp ffiffiffiffiffiffiffi w ðturbulentÞ 5 ðcÞ
183
where
202 203
Rew
g 1 G ¼ tanhw þ 2coshw w coshw The physical rational range of G is worth discussion. To check the sign of G, we have G wg ; hw ¼ 0 for hw ¼ hðcrÞ w , with the critical wedge angle hðcrÞ satisfying w
g 1 tanhðcrÞ 2coshðcrÞ w þ w ¼ 0 w coshðcrÞ w g For hw < hðcrÞ w , G w ; hw < 0, so that Dðg=wÞ < 0. For g hw > hðcrÞ w , G w ; hw > 0, so that Dðg=wÞ > 0. It appears that g=w would be reduced by boundary layer displacement if hw is small and increased if hw is large. For instance, if we set g=w ¼ 0:4, then hðcrÞ w ¼ 142 . However, hw can never be so large, so that practically we have G < 0. Fig. 3 shows that G < 0 is correct for all reasonable hw . When Ma0 ! 1, obli-
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Fig. 3
Value of Gðg=w; hw Þ for various g=w and hw .
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que shock detachment criterion gives the largest hw (45.4°), which restricts the reasonable maximum value of G. This means that the displacement effect of boundary layer reduces g=w.
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2.2. Influence of boundary layer on Mach stem height
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For purpose of clarifying the importance of boundary layer, the magnitudes of the displacement wedge Dhw angle and displacement relative trailing-edge height Dðg=wÞ for a typical range of Reynolds numbers are computed and the sensitivity of the Mach stem height to the wedge angle and trailingedge height is studied. The viscous Mach reflection model is then used to study the viscous effect (Reynolds number effect) on Mach stem height. These studies are performed using the theoretical model and are presented in Sections 2.2.1–2.2.3. The influence of the boundary layer is further studied using CFD, which is also used to verify the accuracy of the theoretical models. The CFD method is presented in Section 3.1. A comparison of inviscid and viscous CFD solutions is given in Section 3.2. The comparison between theory and CFD is displayed in Section 3.3.
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2.2.1. Magnitudes of displacement wedge parameters Fig. 4 displays the displacement wedge angle Dhw computed by Eq. (5) and the relative displacement trailing-edge height Dðg=wÞ computed by Eq. (7). The Reynolds number Rew ¼ Vm00w varies from 105 to 107. The parameters hw ¼ 22 , g=w ¼ 0:4, Ma0 =2.84 and 4.96 are taken from Hornung and Robinson.17 Consider the boundary layer to be laminar. For the effect of Reynolds number, the absolute value of Dhw and Dðg=wÞ increase when Rew decreases or Ma1 increases, due to thickening of the boundary layer by aerodynamic heating at large Mach number. For instance, when Ma0 = 2.84 is fixed and Rew vary from 107 to 105, Dðg=wÞ changes from 0.0018 to 0.0183 and Dhw changes from 0.1 deg to 0.4°. For turbulent boundary layer, the absolute value of Dhw and Dðg=wÞ is smaller than that for laminar flow when Reynolds number is less than 107. For instance, the quantity Dhw 0.12° and Dðg=wÞ 0.0024 for Rew = 107, and increases to Dhw 0.29° and Dðg=wÞ 0.0055 for Rew = 105.
Fig. 4 Influence of Reynolds number on displacement wedge parameters Dhw and Dðg=wÞ for hw ¼ 22 andg=w ¼ 0:4 at Mach numbers Ma1 = 2.84, 4.96.
Though the amount of Dhw appears to be small, its influence on the Mach stem height will be high, as is clear below.
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2.2.2. Mach stem height for various wedge parameters
260
Fig. 5 shows the dependence of the Mach stem height on each factor (hw , g=w), computed by the inviscid Mach refection model of Bai and Wu2, as summarized in Appendix A. To see the influence of hw , we set Ma0 = 2.87, 3.49, 3.98, 4.96 and g=w = 0.4. It is seen that the relative Mach stem height HT =w increases monotonically with hw . When hw changes 1°, HT =w may change by 25% to more than 100%, especially when hw is small. The displacement wedge angle Dhw may be in the order of 1°, as seen from Fig. 4. Hence, it is expected that the viscous boundary layer on the wedge lower surface may have a significant influence on the Mach stem height, especially when hw is small and the Mach number is large. For the influence of the relative trailing-edge height g=w, we set Ma0 = 2.87, 3.49, 3.98, 4.96 and hw ¼ 22 . It is seen that HT =w increases almost linearly with decreasing g=w, for the condition considered here. Thus, the Mach stem height is less sensitive to g=w compared to hw . Thus the influence of the boundary layer on the Mach stem height is majorly due to the increase of the equivalent wedge angle.
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Fig. 5
Mach stem height.
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2.2.3. Effect of Reynolds number
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Now we study the influence of viscous effect, in terms of the Reynolds number, on the Mach stem height using the present viscous Mach reflection model (Section 2.1). The inviscid model of Bai and Wu2 as summarized in Section 2.1 is applied by replacing hw and g=w with the equivalent hðeqÞ and ðg=wÞðeqÞ , w ðeqÞ see Eqs. (3) and (5) for hw , Eqs. (4) and (6) for ðg=wÞðeqÞ . The boundary layer is either assumed fully laminar or fully turbulent, though in real applications the boundary layer may be mixed laminar-turbulent. Fig. 5(b) gives the Mach stem height for Reynolds number Rew ranging from 105 to 107, with hw ¼ 22 , Ma0 =4.96, g=w=0.4. Compared to inviscid mode, the increase of Mach stem height due to viscous effect is large, especially for small Reynolds number and for turbulent boundary layer. For instance, at Rew = 106, the viscous model gives a Mach stem height HT =w0.034 for turbulent boundary layer and HT =w0.033 for laminar boundary layer, compared to HT =w0.029 for inviscid flow. For small Reynolds number, the turbulent model predicts a Mach stem height smaller than
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5
the laminar one, noting that for such small Reynolds number the boundary layer should be laminar. For Rew > 1:04 107 the turbulent model gives a larger Mach stem height, conversely. When Reynolds number tends to infinity, boundary layer thickness tends to zero according to Eq.(B1), and there is no difference between inviscid, laminar and turbulent model’s prediction.
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3. Results analysis
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3.1. Method and boundary conditions for numerical computation
309
For numerical simulation, we use the experimental conditions of Ivanov et al.18 who measured the transition for symmetric reflection in a low-noise wind tunnel. Specifically, we use Ma0 = 4, g=w=0.43 and Rew = 3.83 105 (the Reynolds number based on unit length is 1.278 107). The temperature is T0 = 65 K and the pressure is 1641 Pa. Note that Ivanov et al.18 did not provide these conditions for temperature and pressure and we find these conditions from their previous work. Two wedge angles, hw ¼ 23:21 ; 24:60 , are used for inviscid and viscous computations. For numerical simulations of inviscid flow, the compressible Euler equations in gas dynamics are solved using the second order Roe scheme.19 For viscous flow, the compressible Navier-Stokes equations are solved, with the inviscid part also solved using the second order Roe scheme and with the viscous part solved using the second order central difference scheme. For turbulence modelling in viscous flow computation, the SST two equation model20 is used. All these methods exist in the commercial code Fluent and it is this code that is used for CFD simulation here. The grid used contains 1600 points along the horizontal direction and 700 points along the vertical direction. This grid is much denser than the grid used before by other authors. For instance, Gao and Wu7 used a grid with 300 200 points and Wu et al.21 used grid with 1540 400 points. The grid is refined near the triple point structure and near the wedge lower surface to capture the boundary layer. As displayed in Fig. 2 for boundary conditions, Q1Q2 is a supersonic inlet, AQ1 is a symmetry, Q2Q3 is also a symmetry (reflecting surface), no-slip and adiabatic wall condition is prescribed along AR and RQ₅, Q3Q₄ is a supersonic outlet, and Q₄Q₅ is a nonreflecting boundary. Note that RQ₅ is the backside of the wedge similar to the geometry of Ivanov et al.18 who used a triangular wedge and Q₅ is the upper rear corner of this wedge.
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3.2. Comparison between inviscid and viscous numerical results
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Fig. 6 displays the Mach contours obtained by both inviscid and viscous simulations (the same hw is used). Differences of shock angles are observed for inviscid and viscous flows. The shock angle of the incident shock wave is larger in viscous flow than that in inviscid flow. The Mach stem height in viscous flow is higher than that in inviscid flow, as predicted theoretically above. Shock structures given by current viscous theory are also displayed, and match well with viscous numerical results. Note that there is also a boundary layer along RQ₅ for viscous flow. To see the influence of this boundary layer, we also
346
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Fig. 6 Mach contours by inviscid and viscous flow computations for Ma0 = 4.
Fig. 7 Mach contours by viscous flow computations with RQ₅ treated as wall and non-reflecting boundary for Ma0 = 4 and hw ¼ 23:21 .
Fig. 8 Comparison of Mach contours between the viscous CFD result (with hw ¼ 23:21 ) and modified inviscid CFD result (hw ¼ 23:21 þ Dhw , g=w ¼ 0:43 þ Dðg=wÞ, where Dhw ¼ 0:23 and Dðg=wÞ ¼ 0:0022) for Ma0 = 4.
Fig. 9 Comparison between theory and CFD simulation for Mach stem height at Ma0 = 4, g=w ¼ 0:43 and Rew = 3.83 105.
computed the flow where RQ₅ is replaced by a non-reflecting wall. Fig. 7 displays a comparison of Mach contours with RQ₅ treated as a wall and a non-reflecting boundary, for vis cous flow with hw ¼ 23:21 . It is seen that the Mach stem height is almost the same for both treatments of RQ₅, at least for the conditions considered. It is interesting to see whether the viscous numerical result can be reproduced through inviscid one with correction of the wedge parameters. In Fig. 8, we compare, for Ma0 = 4, hw ¼ 23:21 , viscous CFD result with an inviscid one where hw is augmented by Dhw through Eq.(5) (Dhw ¼ 0:23 in this case) and g/w is changed by Dðg=wÞ through Eq. (6) (Dðg=wÞ ¼ 0:0022 in this case). The close agreement indicates
Table 1 Comparison between theory and numerical simulation for Ma0 = 4, Rew =3.83 105 and g=w ¼ 0:43. The theoretical results are obtained by the model of Bai and Wu (2017)2 for inviscid flow and the present viscous model for viscous flow. Condition
CFD
Theory
Inviscid
Viscous
Inviscid
Viscous
hw ð Þ
p2 =p1
bð Þ
HT =w
bð Þ
HT =w
bð Þ
HT =w
bð Þ
HT =w
23.21 24.60
6.36 6.89
36.23 37.86
0.0878 0.1582
36.77 38.40
0.0989 0.1696
36.24 37.95
0.0870 0.1540
36.52 38.24
0.1021 0.1725
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Fig. 10 Influence of viscous effect on Mach stem height at Ma0 = 4.96, g=w ¼ 0:4 and Rew = 1.36 10⁶.
371
that a simple boundary model correcting the inviscid model is suitable for the present study.
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3.3. Comparison between theory and numerical results
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Table 1 displays the comparison between theory and numerical results for the shock angle and relative Mach stem height. For a given hw and Ma0 , the shock angle b of inviscid flow is computed by the shock angle relation: tanhw ¼
2 tanb
Ma20 sin2 b 1 Ma20 ðc þ cos2bÞ þ
2
For viscous flow, the Eq. (5) is used to find the displacement wedge angle Dhw and then hw is replaced by hw þ Dhw in the above relation to obtain the shock angle in viscous flow. As shown in Table 1, the shock angles obtained using such simplified models match very well with the CFD results. The displacement effect of the boundary layer enlarges the shock angle by an amount close to 0:5 . Good comparison is obtained for the relative Mach stem height HT =w, according to Table 1 and Fig. 9. Fig. 9(a) displays the Mach stem height for hw and shows difference of HT =w at any given hw . Both theory and CFD shows that the Mach stem height is increased by
Fig. 11 model.
Further comparison and validation for current viscous
viscous boundary layer developed on the lower wedge surface. Fig. 9(b), which displays the Mach stem height for the shock angle, shows no difference between inviscid and viscous models, since the displacement effect of the boundary layer is automatically included in the shock angle. The conclusion is similar for other conditions, as displayed in Fig. 10 for Ma0 = 4.96, Rew =1.36 106 g=w ¼ 0:4 and Fig. 11(b) for various Mach and Reynolds number. The condition Ma0 = 3.98 and g=w ¼ 0:4 has been used by various authors for comparison. The comparison of the viscous Mach reflection model with some previous works is displayed in Fig. 11, where we also display the experimental data by Mouton and Hornung,22 the numerical data from inviscid computational fluid dynamics by Mouton and Hornung.6 The predictions by inviscid models of Gao and Wu,7 Mouton and Hornung6 and Bai and Wu2 are also displayed.
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Z. CHEN et al. Table 2
Comparison between theory and previous experimental/numerical results for various conditions.
Condition
Reference result
Theory
hw (°)
g=w
p2 =p1
bð Þ
HT =w
bð Þ
HT =w
2 10 2 106 2 106
23.0 25.0 27.0
0.34 0.34 0.34
8.53 9.68 10.91
33.5 35.8 37.1
0.0493 0.1151 0.2137
33.46 35.91 38.4
0.0649 0.1439 0.2579
3.98 3.98 3.98
1:33 106 1:33 106 1:33 106
21.4 22.3 23.0
0.4 0.4 0.4
5.66 5.98 6.23
34.1 35.12 36.2
0.0271 0.0467 0.0993
34.0 35.2 36.15
0.0269 0.0554 0.0905
3.98 3.98 3.98
1:33 106 1:33 106 1:33 106
22.5 23.0 23.5
0.4 0.4 0.4
6.05 6.23 6.42
35.44 36.03 36.66
0.0642 0.0869 0.1077
35.45 36.04 36.67
0.0658 0.0844 0.1066
Ma0
Rew
Schmisseur (2011)
4.96 4.96 4.96
Mouton and Hornung (2008)22
Mouton and Hornung (2007)6
14
6
408
Comparison between theory and previous experimental/ numerical results for various conditions is shown in Table 2.
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4. Conclusions
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Using a very simple boundary layer model to compute the displacement thickness, we have obtained approximate expressions for the displacement wedge angle and displacement trailing edge height. These displacement parameters are used to correct the geometrical wedge angle and trailing-edge height in the inviscid Mach reflection model of Bai and Wu2. The viscous model thus obtained compares well with the present numerical computation using a very fine grid. The present study shows that the displacement effect of the boundary layer developed over the lower wedge surface has an important effect on the Mach stem height, even though the displacement thickness is very small. It increases the Mach stem height through increasing the effective wedge angle. A slight change of this effective parameter may change, in some conditions, significantly the Mach stem height. Acknowledgement
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This work was supported by the Natural National Science Foundation of China (No. 11802157), Special Foundation of Chinese Postdoctoral Science (No. 2019T120082), and Chinese Post-doc Science Foundation (No. 2018M640119).
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Appendix A. Summary of the inviscid model
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As shown in Fig. 1, the three shock waves and slipline separate the flow into four regions i = 0,1,2,3. In the following we use subscript i to denote solution in region i and superscript T to denote the solution in the vicinity of the triple point. Using the three shock theory of von Neumann gives the shock angles b1 (incident shock wave), bTr (reflected shock wave) and bTm (Mach stem), the Mach numbers Ma1 , MaT2 and MaT3 , the pressures p1 , pT2 and pT3 , and the initial angle of the slipline dTs . These parameters are independent of the Mach stem height. In Eq. (1), the Mach stem height HT should be first guessed and then evaluated through an iterative process. The simplest initial guess is to put HT ¼ g=2.
We choose a coordinate system such that the axis x is along the reflecting surface and the axis y passes the triple point, i.e., the triple point is at (0, HT ), as shown in Fig. 1. For such a coordinate system, the position of the trailing edge R is given by xR ¼ wcoshw ðHA HT Þcotb1 yR ¼ g where HA is the inlet height satisfying HA ¼ g þ wsinhw . The first characteristics of the wedge expansion fan intersects the reflected shock wave at point F. Assuming the shock wave TF to be straight, the position (xF , yF ) can be determined by yF yT ¼ ðxF xT Þtan bTr hw ðA1Þ yF yR ¼ ðxF xR Þtanðl1 þ hw Þ Here l1 ¼ arcsinð1=Ma1 Þ is the Mach angle in region 1. The inviscid model of Bai and Wu2 contains an analytical model for the free part of the slipline, one model for the reflected shock wave and one model for the interacting part of the slipline. The slipline model requires the inlet condition of the quasione-dimensional duct to be specified. The simplest way is to 1 T N N put Mam ¼ 12 MaT3 þ MaN g ,pm ¼ 2 p3 þ pg , where Mag and pN g are normal shock solutions of the Mach number and pressure at the foot of the Mach stem. The shape (height Hs ) of the free part TB of the slipline is given by Hs HT
þ 2ðcþ1 Mam c1Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # Mas c1 c þ 2 #ðMas Þ1 c1 #ðMam ÞN
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cþ1 N 2c
N¼
pT 2 pm
ðA2Þ
c T c1 # Ma2
Maþ s
Here, is the Mach number just above the slipline and is related to the deflection angle of the slipline ds ¼ arctanðdHs =Hs Þ by the Prandtl-Meyer function qffiffiffiffiffiffi ¼ ds dTs þ v MaT2 . Here vðMaÞ ¼ cþ1 arctan v Maþ s c1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c1 ðMa2 1Þ arctan Ma2 1 and c þ 1 #ðMaÞ ¼ 1 þ c1 Ma2 . Eq. (A2) was obtained by pressure bal2 ance between the quasi-one-dimensional duct and secondary expansion waves over the slipline. The interacting part (BE) of the slipline depends on the shape of the reflected shock wave. Now consider the shape of the reflected shock wave. Let hf be the deflection angle of the flow stream in the expansion fan,
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with hf ¼ hw for the leading characteristics RF. When the flow direction deflects downwards, hf is considered positive. For any given hf , the Mach number Mf and the pressure pf in the expansion fan follow from the Prandtl-Meyer solution. The interacting part (segment FK in Fig. 1) of the reflected shock wave has a shock angle br following the expression dbr dhf
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¼
ðbh K5 bM Þbh K4 K3 1þ2bh K4 sinð2br Þ
br ¼ bTr at hf ¼ hw
The abscissa xr on the reflected shock is connected to hf through dxr dhf tan2
499 500 501 502
504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521
¼
1Ma2f #ðMaf Þ
524 525
Ma2f 1
ðhf þlf Þþ1
tanðbr hf Þþtanðhf þlf Þ
ðA4Þ
ðxr xR Þ
where lf is the Mach angle at Maf . The parameters involved in (A4) are defined by 8
1 > bh ¼ 1 K1 þ 2K2 Maf cos2 br > > > > 1 > > bM ¼ ½1 K1 þ 2K2 cos2 br K2 sinð2br Þ > > > > > sin2ðb hf þht Þ > > K1 ¼ sinrð2br Þ > > > > < 2ðcþ1ÞMaf sin2 ðbr hf þht Þ i2 K2 ¼ h > ðc1ÞMa2f sin2 br þ2 > > > > > Maf > > ffi # Maf K5 ¼ pffiffiffiffiffiffiffiffiffiffi > > Ma2f 1 > > > > > > Ma2 sin2 br 1 > > ht ¼ hf arctan 2cotbr 2 f : Starting from point F, where the shock angle is br ¼ bTr , dyr solve Eqs. (A3), (A4) and dx ¼ tan br hf to find r br ¼ br hf and the position of the reflected shock wave (xr , yr ) for hf varying from hw to 0. At any hf or (xr ,yr ), the quantities Mat and pt (solutions just downstream of the reflected shock wave) are connected to Maf and pf through the oblique shock wave expressions, with the shock angle br . The interacting part of the slipline starts at the intersection point B (called turning point since it was found that the slipline changes slope here) of the slipline and the first characteristics of the transmitted expansion waves. The position (xB , yB ) of B can be determined through Eq. (A3) and yB yF ¼ ðxB xF ÞtanðlF þ ht;F Þ where lF ¼ arcsinð1= MaT2 Þ and ht;F ¼ dTs . For the interacting point of the slipline (BE), the deflection angle of the slipline ds satisfies dds cpf Ma2f ðK6 þ K7 Þ þ Wðds Þ1 dxs cþ1
cMa2s p þ s tands ¼ 0 1 Ma2s Hs where
1c2sin2 br 2Ma2f ffi pffiffiffiffiffiffiffiffiffiffi K6 ¼ 2 Maf 1
527 528 529
¼ Wðds Þ
Wðds Þ ¼
ðK6 þK7 Þpf Ma2f
pffiffiffiffiffiffiffiffiffiffi ffi 2 Mat 1
ðcþ1Þpt Ma2t
ðA6Þ
K
531
where K¼
532 533
tan ðlt þ ht Þ þ 1 tanðlt þ ht Þ tands 2
Ma2t
1 þ #ðMat Þ ðxs xG Þ Ma2t 1
535
The parameters with subscript t have been given in the step for the reflected shock wave. The flow parameters below the slipline are related to Hs ¼ Hs ðxÞ through the quasi-onedimensional isentropic flow relations. Starting from point B, and knowing Mat and pt from the solution of the shock/expansion wave interaction algorithm, the slipline/transmitted expansion wave interacting Eqs. (A5) dds and (A6) are used to find xs , ds ¼ ds ðxs Þ, dx and s Hs ¼ Hs ðxs Þ, for any hf varying from hw to 0. This gives one throat position xd , with ds ðxd Þ ¼ 0. Use the quasi-one-dimensional isentropic relation to find the Mach number Mas in the quasi-one-dimensional duct. This gives another throat position xM , with Mas ðxM Þ ¼ Mas ¼ 1. If xd ¼ xM , then the initial choice of the Mach stem height is correct and is said to meet the sonic throat compatibility condition. If xd –xM , then the choice of the Mach stem height xT is incorrect and should be updated until xd ¼ xM . Bai and Wu2 updated this height using the method of bisection. For given Ma0 , g=w and hw , the height of the Mach stem is unique.
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Appendix B. The boundary layer model
555
The displacement thickness of the boundary layer developed over the lower surface of the wedge is used here to account for the viscous effect on Mach reflection. The Mach number may be high enough so that boundary layer is affected by aerodynamic heating. This heating increases the boundary layer thickness in practical application though in many wind tunnel experiments the inflow temperature is very low. For both laminar and turbulent flow, a simple idea to account for aerodynamic heating is to use the reference temperature method. In this method, the formulas for boundary layer parameters are taken from incompressible flow, wherein the thermodynamic and transport properties such as density and viscosity in these formulas are evaluated at some reference temperature T that represents the temperature somewhere inside the boundary layer.15 This method was originally developed by Rubesin and Johnson23 and then advanced by various authors including Eckert,24 Eckert and Tewfik,25 Ott and Anderson,26 Dorrance,27 Herwig28 and Meador and Smart,29 see van Oudheusden30 for a review. In this paper, we assume the boundary layer is purely laminar or purely turbulent. For intermediate Reynolds numbers, the boundary layer may be mixed, i.e., the boundary layer near the leading edge is laminar while after a transition point the boundary layer is turbulent. The case of mixed turbulent boundary layer is not considered in this paper. Here we use l to measure the distance along the lower surface of the edge, starting from the leading edge (A). Various simplified models have been developed for evaluating the displacement thickness d1 ðlÞ of boundary layer (see for instance Schlichting31). Let Rel be the Reynolds number based
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537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554
Maf ðcþcos2br Þþ2
cpt Ma2t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ma2t 1 523
ðA3Þ
dxs dhf
9
K7 ¼
K5 bM Þbh K4 K3 2sinð2br Þ ðbh1þ2b h K4 sinð2br Þ
The abscise xs of the slipline is related to hf by
ðA5Þ
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on the inflow conditions in region (0) and on the distance l. For incompressible flow d1 ðlÞ can be evaluated by 8 1:72l ðlaminarÞ < pffiffiffiffiffi Rel d1 ðlÞ 0:046l :p ffiffiffiffiffi ðturbulentÞ 5 Rel
For compressible flow, the reference temperature is used to evaluate the Reynolds number, in such a way that the displacement thickness is evaluated by 8 1:72l ðlaminarÞ > < pffiffiffiffiffiffiffi ðcÞ Rel d1 ðlÞ ðB1Þ 0:046l > ffiffiffiffiffiffiffi ðturbulentÞ :p 5 ðcÞ Rel
ðcÞ
Here Rel ¼ Rel =c is the Reynolds number based on the reference temperature, and c ¼ TT0 llððTT0 ÞÞ is the ChapmanRobinson number. The temperature-dependent viscosity coefficient l is estimated by the Sutherland law 1:5 Ts þ C T lðTÞ ¼ 1:716 105 Pa s TþC C where Ts ¼ 273:15 K and C ¼ 110:4 K, and the reference temperature T is evaluated using the most popular formula by Eckert.24
T 0:5T0 þ 0:5Tw þ 0:22ðTr T0 Þ
ðB2Þ
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Here Tw is the wall temperature and Tr ¼ T0 ð1þ is the recovery temperature, R is the recovery factor, with R 0:85 for laminar boundary layer and R 0:89 for turbulent boundary layer. In this paper we assume an adiabatic wall so that Tw ¼ Tr .
615
Appendix C. Derivation of hw
610 611 612 613
616 617 619 620 621 622 624 625 626 627
629 630
631
632 633
635
R c1 Ma20 Þ 2
The local effective angle ðeqÞ hloc ðlÞ
ðeqÞ hloc ðlÞ
at l is here defined as
dd1 ðlÞ ¼ hw þ dl
and hwðeqÞ is evaluated by average hðeqÞ ¼ w1 w R w dd1 ðlÞ ðeqÞ 1 hw ¼ hw þ w 0 dl dl and Z 1 w dd1 ðlÞ Dhw ¼ dl w 0 dl
Rw 0
dd1 ðlÞ dl. dl
This yields
ðC1Þ
Introducing the Eq. (B1) in Appendix B for d1 ðlÞ into (B1) gives 8 1:72 ðlaminarÞ < pffiffiffiffiffiffiffi ðcÞ Rew Dhw ðC2Þ 0:046 :p ffiffiffiffiffiffiffi ðturbulentÞ 5 ðcÞ Rew
Here ReðcÞ w is the Reynolds number
ðcÞ Rel
at l ¼ w.
Appendix D. Derivation of ðg=wÞ Now we derive Dðg=wÞ by using the obvious relations 8 ðeqÞ ðeqÞ > < L ¼ w coshw ¼ wcoshw HA gðeqÞ ¼ wðeqÞ sinðhw þ Dhw Þ > : HA g ¼ wsinhw
ðD1Þ
Assuming Dhw to be small relative to hw , the first relation of Eq. (D1) gives coshw w w wDhw tanhw cosðhw þ Dhw Þ
wðeqÞ
ðD2Þ
and the second relation of Eq. (D1) gives ðeqÞ
HA g wðeqÞ
ðeqÞ
HA g wðeqÞ
640 641 642
sinhw þ Dhw coshw
Using the third relation of Eq. (D1) we sinhw ¼ ðHA gÞ=w so that the above relation yields
636 637 638
644
get
645 646 647
HA g þ Dhw coshw w
649
The above relation can be combined with (4) in Section 2.1 to give g HA HA Dhw coshw þ ðeqÞ D w w w
650
which, if Eq.(3) in Section 2.1 is used, becomes g HA Dhw coshw þ Dhw tanhw D w w
655 656
ðD3Þ
Putting HA ¼ g þ wsinhw into Eq. (D3) and noting that coshw þ sinhw tanhw ¼ ðcoshw Þ1 2coshw we get g g 1 tanhw þ D 2coshw Dhw w w coshw Replacing Dhw by Eq. (C2) in the above expression finally gives Eq. (6) in Section 2.1.
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References
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