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Hypersonic transition induced by three isolated roughness elements on a flat plate
Accepted Manuscript
Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate Zhiwei Duan , Zhixiang Xiao PII: DOI: Reference:
S0045-7930(17)30281-5 10.1016/j.compfluid.2017.08.006 CAF 3561
To appear in:
Computers and Fluids
Received date: Revised date: Accepted date:
14 December 2015 6 December 2016 4 August 2017
Please cite this article as: Zhiwei Duan , Zhixiang Xiao , Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate, Computers and Fluids (2017), doi: 10.1016/j.compfluid.2017.08.006
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Highlights Boundary layer transition induced by three roughness elements was investigated. The transition is dominated by three-dimensional shear layer instability. Transition region, aerodynamic force and heat flux were quantitatively analyzed. Ramp element is suitable for triggering hypersonic boundary layer transition.
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Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate Zhiwei Duan1 and Zhixiang Xiao2
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Tsinghua University, Beijing, P. R. China, 100084
Hypersonic boundary layer transition from laminar to turbulent induced by three isolated roughness elements (ramp, diamond, and cylinder) was investigated by direct numerical
simulations based on finite volume formulation with high order MDCD scheme. The effects
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of roughness shape on the transition mechanism, aerodynamic and aero-thermodynamic
performances were quantitatively compared. The transition is dominated by threedimensional shear layer instability and the streamwise transition locations are almost the same for all three roughness elements. The width of turbulent wake, the drag increment and the heat flux are the largest for cylinder. At the same time, the turbulent wake is the
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narrowest and drag and heat flux are the lowest for the ramp.
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Key Words: hypersonic; roughness induced transition; boundary layer
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1. Introduction
Roughness-induced transition (RIT) plays an important role in the aerodynamic and thermodynamic design of
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hypersonic vehicles, as rough surface conditions cause earlier transition than smooth surface conditions. A turbulent boundary layer results in higher heat transfer rate and skin friction compared to a laminar one, which complicates
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the design of thermal protection systems for hypersonic vehicles. Turbulent flow also helps to eliminate the boundary layer separation on the forebody of scramjet air vehicles, such as X-43A and X-51A, as well as enhancing the mixture of fuel and air and combustion in the engine. To ensure that the flow becomes turbulent before the inlet,
1
Post-doctoral research assistant, School of Aerospace Engineering, Tsinghua University
2
Associate Professor, School of Aerospace Engineering, Tsinghua University, [email protected]. 2
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a turbulence generator (TG), mostly comprised of an array of roughness elements, is usually designed to enhance the transition. The ramp, diamond, square, half-sphere, and cylinder are the typical TG elements. There are many factors that influence the effect of RIT
[1]
, such as the geometries of roughness elements (e.g.,
height, width, shape, spacing between two roughness elements,) boundary layer thickness, Reynolds number Mach number[2], angle of attack, wall temperature, freestream turbulence intensity [4][5]
,
[3]
, and so on. Among all the
plays the most important role. To design an ideal TG, the roughness
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geometrical factors, roughness height
[2]
element must trigger the transition from laminar to turbulent as effectively as possible. At the same time, the drag increment and aero-heating rate must be as small as possible. Considering this effect, roughness shape is another important factor as far as TG and transition. [6]
experimentally investigated the shape effects of roughness elements on flow-fields and drag
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Whitehead
characteristics in the hypersonic laminar boundary layer. Common spheres, as well as the triangular prisms, cylinders, pinheads, and two vortex-generators (fin- and wedge-type) were tested and compared; the fin-generator showed less drag with the same frontal area compared to other shapes. Berry et al.[7] experimentally compared five different TGs, mainly including diamond and ramp elements, on a 33.3%-scale Hyper-X forebody model in three
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traditional hypersonic wind tunnels, and found that all the TGs successfully enhanced the transition. Trip 2C (one of the ramp-type TGs) was finally selected for the Hyper-X due to its minimization of entrained vortices within the
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turbulent region and under consideration of the model’s thermal survivability. The RIT mechanisms, drag, and aeroheating rate increment were not reported, however.
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Borg and Schneider [3] tested a 20%-scale X-51A forebody in the Boeing and U.S. Air Force Office of Scientific Research Mach-6 Quiet Tunnel with and without a TG, using ramp and diamond roughness elements. This
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experiment mainly focused on the effect of freestream noise on RIT. They observed the RIT regions of ramp and diamond TGs under both quiet and noisy conditions, and found that under quiet conditions, RIT by diamond was
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about 5.1 cm farther upstream than the ramp under identical conditions. Under noisy conditions, however, RIT by ramp and diamond occurred immediately downstream from the compression corner. Tirtey
[8]
experimentally investigated the transition mechanism of several types of isolated roughness elements
including ramp, cylinder, square, and half-sphere in the hypersonic von Karman Institute (VKI) H3 wind tunnel. Results showed that the flowfields around these three-dimensional (3-D) roughness elements had strong similarities with the generation of various numbers of streamwise vortices. However, the influence of the vortices on the 3
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roughness efficiency as far as enhancing transition was not clear. Based on the flow similarities, Tirtey proposed a universal flow structure/transition model. The experimental mean flow data can also be used to validate transitionmodeling methods. Numerical simulations have become a crucial technique for investigating the RIT mechanism. The most common numerical methods applied in this field are direct numerical simulation (DNS) and linear stability theory (LST).
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Choudhari et al. [9], for instance, conducted a parametric study on flow past several isolated roughness elements in a Mach 3.5 laminar flat plate boundary layer using LST analysis. They found that the streak amplitudes of diamond and cylinder roughness were equivalent to each other, while half-sphere caused lower streak amplitude. In other words, the perturbation by half-sphere roughness element is a little weaker than the other two isolated roughness
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elements. Although the base flows between simulation and experiment are not exactly the same, the results of their study do provide a valuable reference. Van den Dynde and Sandham
[10]
investigated the effects of roughness element shape (smooth bump, flat top,
and ramp) and platform (cylinder, square, and diamond) on transition mechanisms using their own DNS tool. They found that the roughness element platform has a marginal effect on instability growth and transition, while the
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frontal profile shape has a larger effect. Bernardini et al. [2] simulated RIT past a square roughness element also using DNS method, and found that the RIT process was controlled by a Reynolds number based on the momentum deficit
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past the obstacle, which accounts for the effect of roughness shape. In a previous study conducted in our laboratory
[11]
, we found that for an array of ramp elements, the horseshoe
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vortices detached from two ramp elements interact with each other and grow stronger due to counter-rotating effect. This suggests that RIT is enhanced compared to the case of an isolated ramp. In the present study, the effects on RIT
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mechanism in the hypersonic boundary layer according to three isolated roughness elements (ramp, diamond, and cylinder) were explored through DNS methodology based on a finite volume formulation. The drag increment and
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aero-heating rate on the roughness elements and the flat plane were also quantitatively measured and compared. The computational conditions were chosen based on measurements presented by Tirtey et al.
[8]
, with Reynolds number
of 2.6107/m.
The remainder of this paper is organized as follow. Section 2 discusses the numerical strategy, including the governing equations, high-order reconstruction, and code validation. The numerical results for RIT past the three
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isolated roughness elements discussed above are compared and analyzed in Section 3. The final section provides a brief summary and conclusion.
2. Numerical Strategy
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Our in-house code, UNITs (Unsteady NavIer-STokes equation solver), based on a cell-centered finite volume formulation with multi-block structured grids, is used to accurately resolve the unsteady flow. The convective terms are solved using the rotated Roe Riemann solver, through fourth-order MDCD-WENO (minimum dispersion and controllable dissipation-weighted essential non-oscillatory) reconstruction for the characteristic variables. A
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modified, fully implicit, lower-upper symmetric Gauss-Seidel (LU-SGS) with Newton-like sub-iteration in pseudo time is selected as the time marching method when solving the Navier-Stokes (N-S) equation.
2.1 Governing Equations and Numerical Method
The 3-D compressible N-S equation can be written as follows:
(1)
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U Fi Fvi t xi xi
where U, Fi, and Fvi represent the flow variable vector, inviscid flux vector, and viscous flux vector (i=1, 2 3), respectively. They are given as follows:
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ui 0 u u u p 1,i 1 i 1i 1 U u2 ，Fi u2ui p 2,i ，Fvi 2i u3ui p 3,i 3i u3 u q E i ( E p)ui ji j
(2)
where is the fluid density, ui (i=1, 2, 3) are the three velocity components, and variables p and E represent the fluid pressure and total energy. E p / [ ( 1)] ui ui / 2 , where is the specific ratio (1.4, in our case.) For the perfect gas, p=RT, where T is static temperature and R is a constant of 287.
ij is the stress tensor, expressed as follows:
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ui u j x j xi
ij
u k ij xk
(3)
where the molecular viscosity coefficient is determined by Sutherland’s law. is the second viscosity coefficient, typically assumed to be -2/3.
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The heat conduction term is determined by Fourier’s law qi kh T xi , where kh is c p / Pr . The Prandtl number Pr is set to 0.72.
2.2 High-order MDCD Reconstruction
The inviscid fluxes are solved using the rotated Roe Riemann solver combined with fourth-order MDCD-WENO
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reconstruction. MDCD is a linear reconstruction technology developed by Ren et al.
[12], [13]
that can achieve
minimum dispersion and controllable dissipation. In order to ensure shock-capturing properties, it is applied as a linear part of WENO schemes with symmetrical stencils. Details regarding the MDCD-WENO scheme can be found
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in References [12, [13].
2.3 Code Validation
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Two of our previous studies validated the DNS with MDCD-WENO scheme that we utilized in this study [11], [14]. Here, we will discuss some comparisons between our study and an experiment on the supersonic RIT past a
Disturbance Tunnel.
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diamond-shape roughness element, conducted by Kegerise et al.[15] in the NASA Langley Supersonic Low
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The Reynolds number (Re) is 10.8 106/m and Mach number (Ma) is 3.5. The free stream and the wall temperature are 92.55 and 290 K, respectively. The roughness height is about 0.48 times of the local undisturbed
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laminar boundary layer thickness, and it is located 41.2mm downstream of the leading edge of the flat plate. The total computational grids are about 67.2 million, y+wall is around 0.15~0.3, and x+ and z+ are about 2~5. A series of slow acoustic waves with random magnitudes, ranging from 1.0 kHz to 2500 kHz in frequency space,
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The flow structures around and behind the roughness element can be visualized utilizing the Q criterion, which is defined as the second invariant of the velocity gradient tensor [16]: Q
u v u w v w u v u w v w x y x z y z y x z x z y
(4)
As shown in Fig. 1, the characteristic flow structures around isolated roughness elements are the vortex tubes in
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the near wake, and the hair-pin vortices in the downstream region[17] of x>110mm. Between 110mm < x < 135mm, regular hair-pin vortices are generated by base flow deformation due to the instability nonlinear development. In this simulation, the disturbance saturation exhibits at about x=135mm, at which point the regular hair-pin vorticity near
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the symmetrical plane begin to split into more vortices.
Fig. 1: Vortex structures of Ma3.5 diamond element by iso-surface of Q-criterion
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The comparisons of mean mass flux (m/me) and R.M.S. mass flux (m’/mmax) of 104 kHz at x=110.6mm are
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presented in Fig. 2. As expected, a low velocity streak near the symmetrical plane due to the roughness blockage can be observed. At the same time, the high-velocity streaks approach to the wall due to high-speed flow downwash. A series of instantaneous flows at x=110.6 mm were recorded, and the most amplified disturbance (104 kHz) was
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identified through Fast Fourier Transformation(FFT) analysis. The DNS results match the measurements very well. The mass-flux (normalized by corresponding boundary layer edge value) profiles on the symmetric plane after
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the roughness are shown in Fig. 3. The numerical results show reasonable agreement with experiment until x=110.6mm, where the instability growth is nearly linear and flow is laminar. After x= 136.5mm, the disturbances
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saturate and transition happens, the numerical results show a bit of discrepancy with experiment.
Fig. 2: Comparisons of mean mass flux (m/me) and R.M.S. mass flux (m’/mmax) of 104 kHz at x=110.6mm 7
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Fig. 3 The normalized mass-flux (m/me) profiles on symmetric plane after the roughness element
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Fig. 4: Streamwise evolution of peak r.m.s. mass-flux fluctuations on symmetric plane
Figure 4 shows the streamwise evolution of peak r.m.s. mass-flux fluctuations on symmetric plane in the
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diamond roughness element’s wake. The dashed line in the figure denotes the roughness location, the symbols denote the measured data for the most amplified frequency of 100 kHz, and the solid line denotes the simulated peak mass-flux fluctuation of 104 kHz. The disturbance experiences linear growth before x=110.6mm, then the growth rate slows down due to nonlinear effect. The simulation gives a close enough disturbance saturation position with experiment, located at x136.5mm. After saturation, the disturbance begins to decay through breakdown. The very close agreement between experimental data and our simulation confirms that our DNS code is capable of simulating the RIT problem accurately, and the grid scale is small enough for current computation. The slight discrepancy in 8
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the highest amplified frequency may have been caused by the differences of inflow disturbance frequency characteristic between simulation and experiment.
3. Results and Discussions
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The computational setups we established in this study are presented below, followed by a discussion on the grid density and computational domains that were used in our computations. The numerical flow-fields around the ramp are then compared with measurements available in the literature. The flow patterns around three roughness elements, the effects of roughness shape on transition mechanism, aerodynamic and aero-thermodynamic performance are analyzed and compared.
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3.1 Computational Setup
Fig. 5: Sketch of flat plate and roughness elements
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Table 1: Geometrical parameters of ramp and flat plate
Ld
Wp
h
d
60mm
100mm
34.1mm
0.8mm
3.2mm
100
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Lu
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The RIT by three isolated roughness elements (ramp, diamond, and cylinder) mounted on a flat plate was investigated in this study; our process was very similar to experiments conducted by Tirtey et al.
[8]
. The plate used
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for most of the simulations is 60mm long and 34.1mm wide, and the plate used for experiments is 260mm long and 100mm wide. The distance between the roughness and the plate leading edge (Lu) is the same in the simulation and experiment (60mm). The height (h) of the roughness elements is 0.8mm, which is almost equal to the local undisturbed boundary layer thickness without the roughness. For the ramp, the width of the leading edge (d) is 3.2mm, which is four times that of the trailing edge; the ramp angle is 10 degrees. The widths of the diamond and cylinder elements (d) are both 3.2mm. Note that in order to be appropriately comparable with the other two elements, the cylinder roughness in simulation is a little narrower than the experimental cylinder (4mm). The geometric 9
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parameters of these three elements are presented in Fig. 5 and Table 1. The filled circles on the roughness elements represent the original points in the numerical simulations. Three unite Reynolds numbers were adopted in Tirtey’s experiments
[8]
: 9.2106, 1.8107, and 2.6107/m. In
this study, we only tested the largest Re of these. The flow parameters, including the Mach number, density, pressure, temperature, boundary layer thickness, unit Reynolds number, and Reynolds number based on the height
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(Reh =uh /) are listed in Table 2. The quiet inflow boundary condition was adopted for all the cases in our simulations, so the boundary layer transition was merely induced by the instability created by roughness elements themselves.
The unit length was 1 meter and the freestream velocity (U) was taken as the referenced velocity. The non-
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dimensional time step was taken as 0.0001, corresponding to 9.937910-8 second. To achieve second order in time marching, eight sub-iterations were applied to ensure the inner residuals decrease 2 to 3 orders. Ten flow-through times, defined as the ratio of the streamwise length over the free-stream velocity, were taken to compute and average. It took approximately 50,000 CPU hours to finish one simulation based on the Fine mesh. Table 2: Flow parameters
p(Pa)
0.0122
2449.42
T(K)
Tw(K)
(mm)
Re (m-1)
Reh
70
300
0.79
2.6107
1.6104
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6
(kg/m3)
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Ma
3.2 Effects of Grid Scale for Ramp Case
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In this subsection, three sets of grids, “Coarse”, “Medium”, and “Fine”, were adopted to compare the effects of
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grid density. For all the grids, the first grid point off the wall is about 10µm, corresponding to y+ 0.15~0.78 and about 100 grid points are distributed in the normal (y) direction with most of them in the boundary layer. In the downstream wake region, the grid scales for the Medium grid are 50µm in the streamwise (x) and
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spanwise (z) directions, corresponding to x+ and z+ 0.75~3.9. This grid density is comparable to the typical mesh adopted in similar DNS studies on RIT, such as those by Bartkowicz [18], Iyer et al.[19] and Bernardini et al.[20]. The primary difference between the Medium and the Coarse grids is the grid scale in the core flow region, where the latter is two times that of the former in streamwise and spanwise directions. In fact, the grid scales of the Fine and Medium grid are almost the same. The core flow region of Fine grid is about 4 times wider in the spanwise direction, in order to cover the wider wake of diamond and cylinder roughness. 10
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To save computational resource, the Fine grid only extends to 100mm downstream of the ramp, while Coarse and Medium grids extend to 230mm. The total grid numbers of Coarse, Medium, and Fine are 20, 27, and 50.7 million,
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respectively. The typical grids around the ramp are presented in Fig. 6 and grid parameters are listed in Table 3.
Fig. 6: Near-field grids around ramp roughness element
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Table 3: Grid parameters in wake core region for different grids
x+, z+
Downstream length (mm)
Total grid number (million)
Coarse
0.15~ 0.78
1.5~7.8
230
20
Medium
0.15~ 0.78
0.75~3.9
230
27
Fine
0.15~ 0.78
0.75~3.9
100
50.7
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y+wall
Figure 7 shows the skin friction coefficients (Cf) in the wake based on the three grids, which are nearly the same
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before x=10mm where the flow is still laminar. Between 10 and 100mm, in which region flow transition occurs, the Cf coefficients gradually increases to peak value, then decreases to a theoretical turbulent Cf value on a flat plate. The differences in Cf indicate that the flow transition occurs slightly downstream, about 10 mm, when simulating
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with the Coarse grid. The Cf coefficients by Medium and Fine grids are practically identical. In the further
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downstream region, Cf coefficients by Coarse, Medium, and Fine grids approach the theoretical turbulent skin friction value on the flat plate. Comparisons of five streamwise velocity profiles at x=10, 20, 30, 40, 50mm on the symmetric plane (z=0) and
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on the plane of z= 0.4mm are presented in Fig. 8. In the linear and weakly nonlinear disturbance growth region, where x is less than 40mm, the differences among the three grids are very small. At x=50mm, the nonlinear disturbance growth is dominated and the difference between the Coarse and Medium grids is more prominent than that between Medium and Fine grids. It indicates that the grid convergence is achieved and the scale of Fine grid is sufficient to resolve the transitional flow in the wake of roughness elements.
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Recall that the Fine grid density is better than that of some typical DNS simulation
[18][19][20]
. In this study, the
transition region and tendency are almost the same for the three grids, except that the transition location is a little downstream for the Coarse grid. For this reason, we assert that Medium and Fine grid scales are small enough to resolve the RIT problem. The following discussions on flow features and the transition mechanism are based on the
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grid scale of the Fine grid.
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Fig. 7: Cf distributions in wake region on the symmetric plane
Fig. 8 Streamwise velocity profiles on the symmetric plane (top row) and z=0.4mm plane (bottom row)
3.3 Qualitative Comparison with Experiment for Ramp Case
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In the experiment
[8]
, infrared thermography measurement, oil, and Acenaphthene sublimation visualization
techniques were adopted to depict the transition procedure after the roughness element. Infrared thermography and sublimation techniques can be used to highlight the high heat flux region, and oil visualization applied to depict the high skin friction region. As we know, the convective heat-flux Qw to the wall is significantly higher in turbulent boundary layer than
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laminar boundary layer, so Qw is typically used to judge the laminar or turbulent state of the flow. The nondimensional heat-flux rate is the Stanton number. We adopted the modified Stanton number for convenience, where freestream density and velocity were used instead of the boundary layer edge conditions, and total temperature was used to replace adiabatic wall temperature. The modified Stanton number is defined as follows:
Qw u cp (T0 Tw )
(5)
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St
For the ramp case, comparisons between Stanton numbers and skin friction coefficients in the wake for computations and measurements are shown in Fig. 9.
The computational flow topological structure is the same as that of the experiment in the vicinity of the
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roughness element. Two high heat-flux rate and skin friction streaks exist due to the vortex tube, then the two streaks combine into a single steak between x=40 and 80mm in the simulation, where the flow transition begins.
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Near the outflow, the two streaks spread out in the spanwise direction. It seems that the vortex tube near the roughness is a little shorter in the experiment, and near the outflow, the
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disturbed flow region is a little wider than the computational result. It should be noted that the experiment was conducted in a noisy wind tunnel, but the quiet inflow condition was applied in the DNS study. Differences in
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freestream conditions may have caused some deviation in flow topology.
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Fig. 9: Comparisons of Stanton number St and skin friction coefficient Cf distributions (Top: Simulation; Bottom: Measurements by infrared thermography and oil-visualization)
3.4 Mean and Instantaneous Flow-fields Around the Three Roughness Elements Because diamond and cylinder geometries differ from that of the ramp, the grid distributions around them
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also differ, as shown in Fig. 10. The grid scales in the wake are the same as the Fine grid for the ramp.
Fig. 10: Grids around isolated diamond and cylinder roughness elements
Figures 11 and 12 present the mean flow patterns in the near-field of roughness elements. Any roughness element in the boundary layer will generate an adverse pressure gradient, which may cause separation before it. For
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the ramp, a single upstream separation bubble is observed. The adverse pressure gradient before the diamond is a little stronger, and two pairs of separation bubbles are observed. As the bluntness of the cylinder is the largest
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among three elements, its resultant adverse pressure gradient is the largest, generating three pairs of unstable separation bubbles. The flow diverges very rapidly after the roughness element, forming the low-pressure region,
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where a main recirculation can be easily observed. In hypersonic or supersonic boundary layers, a shock wave forms before the roughness element due to
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supersonic/hypersonic flow. For the cylinder, an obvious separation shock forms before the main separation bubble
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near the roughness element, similar to cases with higher cylinders [5][14][18].
Fig. 11: Mean streamlines and pressure distribution on symmetric plane (from left to right: ramp, diamond, cylinder)
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According to the comparisons of surface streamlines, the horse-shoe vortices past the square and cylinder are much stronger and wider than that of ramp. At the same time, the Stanton numbers at the nose region are also much
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larger than that of ramp.
Fig. 12 Surface streamlines and contours of Stanton number around the three roughness elements
As shown in Fig. 11, a strong “bow shock” is observed before the roughness element in the hypersonic boundary
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layer. At the same time, a series of separation shocks presents due to the separation bubbles. Expansion waves and reattachment shock waves are formed after the roughness element.
Shocks should be plotted clearly, despite the fact that they are very complex and difficult to fully demonstrate. Fortunately, they can be fairly readily illustrated using the “normal Mach number” parameter (Man)
[21]
. The iso-
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surface of Man=1 stands for the shock wave, defined as follows:
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(6)
Fig. 13: Shock wave around roughness elements
As shown in Fig. 13, for the ramp, the upstream separation bubble is very small and the separation shock is
restricted to a narrow region along the spanwise direction. Both the separation and reattachment shocks are convex. For the diamond, the separation shock is the widest due to the largest upstream separation flow (Fig. 12 and Fig. 14) and the downstream reattach shock exhibits a concave shape. The shock system for the cylinder is similar with those
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past the diamond, apart from the flat reattachment shock. The separation shock and bow shock join to form a single shock surface at 1.53, 3.17 and 3.83 mm downstream, for ramp, diamond and cylinder, respectively. The separation bubbles around these three roughness elements are illustrated below using the iso-surface of a small negative streamwise velocity (-110-4 U), colored by the streamwise Reynolds stress u’u’ , as shown inFig. 14. For the ramp, the upstream separation flow is restricted to a relatively small zone about 1.98mm long and
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2.32mm wide. The small u’u’ on the separation flow surface indicates that the separation may be quite steady. After the ramp, the separation bubble is about 3.72mm long, and larger u’u’ is observed.
For the diamond, the separation extends about 3.91 upstream and 11.22mm in the spanwise direction. The u’u’ is small near the symmetric plane, but gradually increases in magnitude as moving away from the symmetric plane.
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The downstream separation is about 7.6mm long, and its u’u’ is at a higher level than that at the upstream separation flow. For the cylinder, the upstream separation is the longest (about 5.25mm) among the three roughness elements, and 6.85mm wide. The u’u’ before the cylinder is about two orders higher than that past the diamond, and the
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downstream recirculation is the shortest, about 3.02mm in the streamwise direction.
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Fig. 14: Mean separation bubbles around three roughness elements
Fig. 15: Mean vortices near roughness elements colored by streamwise velocity and coefficients of skin friction
The vortices in space around the roughness elements are shown in Fig. 15. With increasing bluntness of the roughness element, the adverse pressure gradient formed on the windward side of the obstacle also increases, 16
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pushing the flow farther away from the elements. The wakes of the diamond and cylinder are wider than that of the ramp due to the stronger horseshoe vortices; the horseshoe vortices of the diamond reach a farthest location in the
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spanwise direction.
Fig. 16: Instantaneous non-dimensional temperature at sections z/d=0 and y/h=0.5
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(from top to bottom: ramp, diamond and cylinder, respectively)
Flow structures at the spanwise (z/d=0) and normal (y/h=0.5) sections are shown by temperature contours in Fig.
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16. The roughness element in the boundary layer slows down the main flow and several low-velocity and high temperature streaks are then observed behind the roughness. The high-velocity flow out of the boundary layer and
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the low-velocity flow in the wake form a very strong 3-D shear layer near the symmetric plane, which is very stable near the roughness before x=35mm. In the regions further downstream (from 35 to 60mm) some regular structures
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arise, indicating the nonlinear instability breakdown. In the region approaching the outlet (about from 60 to 90mm) the regular structures gradually develop, evolve, and split into many small-scale and chaotic structures. The similar
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phenomena were likewise observed in studies on cylindrical [5][18] and hemispherical roughness elements [19]. In the y/h=0.5 section, the most distinct structures are the horseshoe and wake vortices. For the ramp, almost no
horseshoe vortices are observed; for the diamond, the horseshoe vortices are very stable. For the cylinder, these horseshoe vortices are unstable due to strong disturbance from the cylinder, and nonlinear instability breakdown starts at about x=20mm (about 15 mm farther upstream than those of the shear layer on the symmetric plane.) At the
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end of the computation domain (100mm,) the width of the disturbed wake for the cylinder is the widest, about 16mm; for the ramp and diamond, they are about 8 and 12mm, respectively. As shown in Fig. 17, the most distinct structures are the stable wake vortex pairs from both sides of the roughness just beside the symmetric plane. For diamond and cylinder roughness elements, horseshoe vortices are especially readily observed. The wake vortices grow weaker as moving downstream until break down into many
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nearly regular, coherent hairpin vortices, marking where transition occurs. Between the hairpin vortices and the flat plate, a few small-scale streamwise vortices can also be found. The horseshoe vortices of the diamond are farther from the mid-plane than those of the cylinder. And for the ramp, the horseshoe vortices are much closer to the
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roughness and more difficult to distinguish.
Fig. 17: Instantaneous vortex structures using iso-surface of Q criterion colored by streamwise velocity (from top to bottom: ramp, diamond, cylinder)
Unsteady pressure fluctuations on the flat plate in the separation zone before and behind the roughness were
recorded as shown in Fig. 18. For the ramp, the sample point before the roughness was located at (-4.5 0, 0mm) and the sample point after the roughness was located at (2.3, 0, 0mm). For the diamond and cylinder, the upstream and downstream sample points were the same, (-2.4, 0, 0mm) and (3.4, 0, 0mm), respectively. According to comparative 18
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analysis of power spectral density (PSD), the upstream and downstream separation bubbles show some distinct periodic motions; the dominating frequencies are about 55.27, 30.19, and 42.58 kHz for the ramp, diamond, and cylinder, respectively. For the ramp, the unsteady flow exhibits several notable harmonic frequencies, including those at 110.54, 165.82, and 221.09 kHz. For the diamond and cylinder, however, there are no such harmonic
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frequencies. The magnitude of fluctuation energy for the ramp is the smallest, while that of cylinder is the largest.
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Fig. 18: Normalized pressure fluctuation before and after roughness elements
3.5 Effects on Transition
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Figure 19 shows the streamwise Reynolds stress (u’u’) distribution at four streamwise sections (x=10, 40, 70, and 100 mm, respectively) in the roughness wake. The contour lines of us
u
y u z , which reflect the 2
2
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local shear level, are also plotted in the same figure. By comparing information shown in Fig. 15 and Fig. 19, we found that us distributes around the low-velocity
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streak forming 3-D high shear level vortices. These 3-D shear layers show unstable characteristics, and the Reynolds stress distributes right around their boundaries.
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For the ramp, the disturbed region is narrow at x=10mm section. At x=40mm section, the instability of the wake
vortices grows larger and the horseshoe vortices near the flat plane become less stable. At x=70mm section, the horseshoe vortices are less stable than the wake vortices, and the instability zone becomes wider. At the outlet of the computational domain (x=100mm), the half width of the fully turbulent boundary layer is about 1mm. When z is greater than 1mm, the flow remains in the transitional state.
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For the diamond and cylinder, the horseshoe vortices are less stable than the wake vortices at x=10mm section, and the flow is constrained in the low-speed streak region. For the diamond, the instability of the wake vortices grows faster than that of the horseshoe vortices; for the cylinder, both the wake vortices and the horseshoe vortices play important roles in the flow transition. At the end of the computation domain (x=100mm), the half width of the
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fully turbulent wake is about 2.5mm for the diamond and about 6mm for the cylinder.
Fig. 19: Contours of streamwise Reynolds stress and line of us at four streamwise sections
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(from left to right: ramp, diamond, cylinder)
Fig. 20: TKE Contours in wake of roughness at y=0.4mm and Prms on flat plate (from top to bottom: ramp, diamond, cylinder) 20
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The lateral spread of the turbulent wedge is shown in Fig. 20, through TKE (turbulent kinetic energy) distribution on the y/h =0.5 (y=0.4mm) plane and the Prms (root mean square of pressure) distribution on the flat plate. The shapes of the transitional/turbulent wake are similar among all three isolated roughness elements. For the ramp, the transition begins at about 40mm near the symmetric plane, and for the diamond, the transition starts a little further downstream, (about 50mm,) also near the symmetric plane.
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For the cylinder, the flow transition is much more complex than those of the ramp and diamond. Near the symmetric plane, the transition occurs at about x=25mm. However, the flow reaches the transitional/turbulent state immediately after the roughness, due to the unsteady horseshoe vortices.
At the outlet of the computational domain (x=100mm), the half width of the transitional/turbulent wedge of the
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ramp is about 4mm, and about 6mm and 8mm for the diamond and cylinder, respectively.
Fig. 21: Streamwise TKE evolution
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(Left: Line integration on the symmetric plane; Right: Surface integration on the y-z plane)
The streamwise TKE evolutions are presented in Fig. 21. Two kinds of integration are applied: one is the line
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integration in the boundary layer, defined by TKELine = ò
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normal and spanwise directions, defined by TKESurface = ò
y=d y=0
ui¢ui¢ 2 dy , another is the surface integration in both
z=zmax z=zmin
ò
y=d y=0
ui¢ui¢ 2 dydz . It should be noted that the vertical
axis is in logarithmic form.
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The TKELine mainly represents the disturbance evolution on the symmetric plane, transported by the wake
vortices. A significantly linear growth rate is observed from 15 to 55mm for cylinder, from 15 to 40mm for ramp, and from 30 to 60mm for diamond, respectively. From 20 to 40mm, the TKEline of cylinder roughness is the largest, while its growth rate is the smallest. The growth rate of ramp is the largest with medium TKEline. The TKEline of diamond is the smallest with medium growth rate. The TKEline growth of ramp slows down at about x=40mm, but it happens at about x=60mm for both the diamond and cylinder. The saturation is achieved at about x=80mm for the 21
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ramp, where transition is finished. For cylinder and square, saturations are achieved at about x=70mm. When x is greater than 80mm, the TKEs of ramp, diamond, and cylinder are nearly the same. It indicates that the flow is fully turbulent. The TKESurface is the total TKE on the y-z plane, transported by wake and horseshoe vortices simultaneously. The curves exhibit the same disturbance linear growth region and tendency with TKEline results. The TKESurface of
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cylinder is the largest in the whole computational domain. It indicates that the disturbances by horseshoe vortices play a key role. The TKESurface of ramp is smaller than that of diamond before x=20mm, while it grows with the highest linear growth rate after x=20mm.
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3.6 Effects on Aerothermodynamics and Aerodynamic Performance
An ideal roughness element will trip the flow from laminar to turbulence very rapidly without producing much drag at the roughness itself or encountering severe aerothermal load (high temperature or aero-heating rate). We investigated not only the transition effects of the three roughness elements in our numerical simulation, but also the
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aerothermal load and drag increment caused by the roughness element.
Fig. 22: Stanton number on wall (from top to bottom: ramp, diamond and cylinder)
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Figure 22 presents the non-dimensional heat flux (Stanton number) distribution on the roughness and flat plate, where the computational flow topological structures are same as that of the experiment[8]. In the vicinity of the roughness elements, two high heat-flux rate streaks exist due to the steady vortex tube. For the ramp, the two streaks combine into a single steak between 40 and 80mm, where the flow transition begins. Near the outflow, the heat flux streak spreads in the spanwise direction. For the diamond, the two heat flux streaks near the symmetric plane
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separate, and the convective heat is higher than that of the ramp. Two high heat flux streaks are formed at the side region (z5.5mm) due to the horseshoe vortices, identical to the instantaneous temperature distribution shown in Fig. 16. For the cylinder, three pairs of separation bubbles form and wrap around the element. As a result, six clearly distinguishable high heat flux streaks are formed in the wake. The distributions of the heat flux are almost uniform
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after x80mm in the spanwise direction for all the three roughness elements.
The widths of the transitional/turbulent wake are about 8, 12, and 16mm for the ramp, diamond, and cylinder, respectively, coincident with the temperature and TKE distributions shown in Fig. 16 and Fig. 20. The temperature and pressure dramatically increase after the strong bow shock near the head of each roughness element. As a result, the roughness itself will encounter a severe aerothermal load. As shown in Fig. 12 and Fig. 22,
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the heat flux around the top of the roughness is massive compared to the nearby laminar zone. The value and
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location of the maximum heat flux on the roughness element surfaces are listed in Table 4. For the ramp, maximum heat flux at the top surface is 16.691 times of local undisturbed laminar heat value. For the diamond and cylinder,
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the maximum heat flux magnitudes are 133.691 and 173.793 times of laminar heat value，located at their windward vertices.
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Table 4: Maximum non-dimensional heat flux (103) for roughness elements Hmax
Hlam
Hmax/Hlam
Position
Ramp
4.106
0.246
16.691
Top
Diamond
32.888
0.246
133.691
Windward vertex
Cylinder
42.753
0.246
173.793
Windward vertex
Comparisons among the normalized heat flux at three spanwise sections are presented in Fig. 23. The solid, dashed, and dash-dot-dotted lines represent the ramp, diamond, and cylinder, respectively. On the symmetric plane 23
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(z/d=0), the heat flux of the ramp wake is larger than that of the diamond or cylinder from 10 to 80mm, reaching a peak (St1.510-3) at 65mm. When x is larger than 80mm, the heat fluxes of the three roughness elements reach a plateau around St=0.810-3, about 5~6 times that of the local laminar value, representing the typical level of turbulent flow. At sections of z/d=0.5 and 1.0, the heat flux of the ramp wake is the smallest due to its narrow
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transitional/turbulent wake, while the high heat flux streak of the cylinder is the widest.
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Fig. 23: Normalized heat flux along streamwise direction on three spanwise positions Table 5: Drag coefficients (104) for roughness elements Cd (t)
Cd (r)
Cd (f)
Cd_p (r)
Cd_v (r)
Ramp
8.449
0.124
8.325
0.104
0.020
Diamond
9.280
0.640
8.640
0.552
0.089
Cylinder
9.863
0.595
9.267
0.489
0.100
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The drag caused by the roughness was also evaluated as listed in Table 5. The referenced area is the area of flat plate (5.45610-3 m2). Cd (t), Cd (r), and Cd (f) represent the total drag, the drag by the roughness itself, and the drag of the flat plate, respectively, and Cd_p and Cd_v stand for pressure and viscous drag. The drag force by the roughness elements themselves are rather small compared to the overall drag, about 1.47,
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6.90, and 6.03% of the total drag for the ramp, diamond, and cylinder, respectively. Most of the total drag is comprised of the viscous drag on the flat plate, which is affected by the transitional/turbulent wake. For the roughness elements themselves, the drag force by diamond and cylinder are similar and much larger (about 5 times)
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than that provided by the ramp.
4. Conclusions
The hypersonic boundary layer transition induced by three types of isolated roughness element (ramp, diamond, and cylinder) has been explored in this study using the DNS method. The width and height of the three roughness
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elements are the same, and their heights approximately equal to the local undisturbed boundary layer. The flow transition is dominated by the instability of the counter-rotating streamwise vortices generated by the
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roughness. For the cylinder, the horseshoe vortices play a more important role than the wake vortices just behind the roughness. The transition distance downstream of the ramp is about 80mm, a little longer than that of the diamond or
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cylinder (70 mm,) and the width of the transitional/turbulent wake of the ramp is smaller than those of the other two obstacles, but the growth rate of the disturbance for the ramp is the largest. Conversely, the maximum heat flux and
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drag increment caused by the ramp are the smallest among the three roughness elements, while they are the largest for the cylinder.
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An ideal roughness element should trip the flow from laminar to turbulence very rapidly, without producing
much drag at the roughness itself or encountering severe aerothermal load. We posit that the ramp is a suitable roughness element for triggering hypersonic boundary layer transitions when accounting for transition effect, aerothermal load, and the drag increment among the three roughness elements we investigated.
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Acknowledgments This work was supported by the National Science Foundation of China (Grant number 11372159) and China Postdoctoral Science Foundation (Grant number 2015M571029). The authors thank Tsinghua National Laboratory
References
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for Information Science and Technology for the computational resources.
[1] Schneider, S.P, “Effects of Roughness on Hypersonic Boundary-Layer Transition,” Journal of Spacecraft and Rocket, Vol.45, No. 2, Mar.-April 2008, pp.192-209; doi: 10.2514/1.29713
[2] Bernardini M., Pirozzoli S. and Orlandi Paolo., “Parameterization of Boundary Layer Transition Induced by Isolated Roughness Elements,” AIAA Journal, Vol. 52, NO. 10, October 2014, pp. 2261-2269;
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[3] Matthew P. Borg and Steven P. Schneider. Effect of Freestream Noise on Roughness-Induced Transition for the X-51A Forebody, J. of Spacecrafts and Rockets. 45(6), pp. 1106-1117, 2008. doi: 10.2514/1.38005
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[6] Whitehead, A. H., Jr., “Flowfield and Drag Characteristics of Several Boundary-Layer Tripping Elements in Hypersonic Flow,” NASA, TND-5454, Oct. 1969
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[Phd thesis]. Von Karman Institute for Fluid Dynamics, Oct. 2008; [9] Choudhari,M., Li,F., Wu,M., Chang, C.L., Edwards,J., Kegerise, M. and King,R., “Laminar-Turbulent Transition behind Discrete Roughness Elements in a High-Speed Boundary Layer, ” AIAA paper 2010-1575, Jan. 2010;
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[10] Van den Dynde, J.P.J.P. and Sandham, N.D., “Numerical Simulation of Rouhgness Induced Instability Growth and Transition at Mach 6,” AIAA paper 2014-2499, June 2014;
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[12] Sun, Z.S., Ren, Y.X., Larricq, C., Zhang, S.Y., and Yang, Y.C., “A Class of Finite Difference Schemes with Low Dispersion and Controllable Dissipation for DNS of Compressible Turbulence,” Journal of computational physics, Vol. 230, Issue 12, June 2011, pp.4616-4635; doi: 10.1016/j.jcp.2011.02.038 26
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[13] Wang, Q.J., Ren, Y.X. and Sun, Z.S., “High Resolution Finite Volume Scheme Based on Minimized Dispersion and Controllable Dissipation Reconstruction,” ICCFD7-1004, July 2012; [14] Duan, Z.W., Xiao, Z.X. and Fu, S., “Direct Numerical Simulation of Hypersonic Transition Induced by an Isolated Cylindrical Roughness Element,” Science China Physics, Mechanics & Astronomy, Vol. 57, NO.12, pp. 2330-2345; doi:10.1007/s11433-014-5556-4 [15] Kegerise, M.A., King, R.A., Choudhari, M. Li, F., and Norris, A., “An experimental study of roughness-induced instabilities in a supersonic boundary layer,” AIAA paper 2014-2501, June, 2014;
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Minnesota, 2012;
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[18] Bartkowicz, M. D., “Numerical simulations of hypersonic boundary layer transition,” [Phd thesis], University of [19] Iyer, P.S. and Mahesh, K., “High-speed Boundary Layer Transition Induced by a Discrete Roughness Element,” Journal of Fluid Mechanics, Vol.729, 2013, pp.524-562; doi:10.1017/jfm.2013.311
[20] Bernardini, M., Pirozzoli, S. and Orlandi, P., “Compressibility Effects on Roughness-Induced Boundary Layer Transition,” International Journal of Heat and Fluids Flow, Vol. 35, June 2012, pp.45-51; doi: 10.1016/j.ijheatfluidflow.2012.02.007
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Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate Zhiwei Duan , Zhixiang Xiao PII: DOI: Reference:
S0045-7930(17)30281-5 10.1016/j.compfluid.2017.08.006 CAF 3561
To appear in:
Computers and Fluids
Received date: Revised date: Accepted date:
14 December 2015 6 December 2016 4 August 2017
Please cite this article as: Zhiwei Duan , Zhixiang Xiao , Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate, Computers and Fluids (2017), doi: 10.1016/j.compfluid.2017.08.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights Boundary layer transition induced by three roughness elements was investigated. The transition is dominated by three-dimensional shear layer instability. Transition region, aerodynamic force and heat flux were quantitatively analyzed. Ramp element is suitable for triggering hypersonic boundary layer transition.
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Hypersonic Transition Induced by Three Isolated Roughness Elements on a Flat Plate Zhiwei Duan1 and Zhixiang Xiao2
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Tsinghua University, Beijing, P. R. China, 100084
Hypersonic boundary layer transition from laminar to turbulent induced by three isolated roughness elements (ramp, diamond, and cylinder) was investigated by direct numerical
simulations based on finite volume formulation with high order MDCD scheme. The effects
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of roughness shape on the transition mechanism, aerodynamic and aero-thermodynamic
performances were quantitatively compared. The transition is dominated by threedimensional shear layer instability and the streamwise transition locations are almost the same for all three roughness elements. The width of turbulent wake, the drag increment and the heat flux are the largest for cylinder. At the same time, the turbulent wake is the
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narrowest and drag and heat flux are the lowest for the ramp.
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Key Words: hypersonic; roughness induced transition; boundary layer
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1. Introduction
Roughness-induced transition (RIT) plays an important role in the aerodynamic and thermodynamic design of
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hypersonic vehicles, as rough surface conditions cause earlier transition than smooth surface conditions. A turbulent boundary layer results in higher heat transfer rate and skin friction compared to a laminar one, which complicates
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the design of thermal protection systems for hypersonic vehicles. Turbulent flow also helps to eliminate the boundary layer separation on the forebody of scramjet air vehicles, such as X-43A and X-51A, as well as enhancing the mixture of fuel and air and combustion in the engine. To ensure that the flow becomes turbulent before the inlet,
1
Post-doctoral research assistant, School of Aerospace Engineering, Tsinghua University
2
Associate Professor, School of Aerospace Engineering, Tsinghua University, [email protected]. 2
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a turbulence generator (TG), mostly comprised of an array of roughness elements, is usually designed to enhance the transition. The ramp, diamond, square, half-sphere, and cylinder are the typical TG elements. There are many factors that influence the effect of RIT
[1]
, such as the geometries of roughness elements (e.g.,
height, width, shape, spacing between two roughness elements,) boundary layer thickness, Reynolds number Mach number[2], angle of attack, wall temperature, freestream turbulence intensity [4][5]
,
[3]
, and so on. Among all the
plays the most important role. To design an ideal TG, the roughness
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geometrical factors, roughness height
[2]
element must trigger the transition from laminar to turbulent as effectively as possible. At the same time, the drag increment and aero-heating rate must be as small as possible. Considering this effect, roughness shape is another important factor as far as TG and transition. [6]
experimentally investigated the shape effects of roughness elements on flow-fields and drag
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Whitehead
characteristics in the hypersonic laminar boundary layer. Common spheres, as well as the triangular prisms, cylinders, pinheads, and two vortex-generators (fin- and wedge-type) were tested and compared; the fin-generator showed less drag with the same frontal area compared to other shapes. Berry et al.[7] experimentally compared five different TGs, mainly including diamond and ramp elements, on a 33.3%-scale Hyper-X forebody model in three
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traditional hypersonic wind tunnels, and found that all the TGs successfully enhanced the transition. Trip 2C (one of the ramp-type TGs) was finally selected for the Hyper-X due to its minimization of entrained vortices within the
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turbulent region and under consideration of the model’s thermal survivability. The RIT mechanisms, drag, and aeroheating rate increment were not reported, however.
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Borg and Schneider [3] tested a 20%-scale X-51A forebody in the Boeing and U.S. Air Force Office of Scientific Research Mach-6 Quiet Tunnel with and without a TG, using ramp and diamond roughness elements. This
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experiment mainly focused on the effect of freestream noise on RIT. They observed the RIT regions of ramp and diamond TGs under both quiet and noisy conditions, and found that under quiet conditions, RIT by diamond was
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about 5.1 cm farther upstream than the ramp under identical conditions. Under noisy conditions, however, RIT by ramp and diamond occurred immediately downstream from the compression corner. Tirtey
[8]
experimentally investigated the transition mechanism of several types of isolated roughness elements
including ramp, cylinder, square, and half-sphere in the hypersonic von Karman Institute (VKI) H3 wind tunnel. Results showed that the flowfields around these three-dimensional (3-D) roughness elements had strong similarities with the generation of various numbers of streamwise vortices. However, the influence of the vortices on the 3
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roughness efficiency as far as enhancing transition was not clear. Based on the flow similarities, Tirtey proposed a universal flow structure/transition model. The experimental mean flow data can also be used to validate transitionmodeling methods. Numerical simulations have become a crucial technique for investigating the RIT mechanism. The most common numerical methods applied in this field are direct numerical simulation (DNS) and linear stability theory (LST).
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Choudhari et al. [9], for instance, conducted a parametric study on flow past several isolated roughness elements in a Mach 3.5 laminar flat plate boundary layer using LST analysis. They found that the streak amplitudes of diamond and cylinder roughness were equivalent to each other, while half-sphere caused lower streak amplitude. In other words, the perturbation by half-sphere roughness element is a little weaker than the other two isolated roughness
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elements. Although the base flows between simulation and experiment are not exactly the same, the results of their study do provide a valuable reference. Van den Dynde and Sandham
[10]
investigated the effects of roughness element shape (smooth bump, flat top,
and ramp) and platform (cylinder, square, and diamond) on transition mechanisms using their own DNS tool. They found that the roughness element platform has a marginal effect on instability growth and transition, while the
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frontal profile shape has a larger effect. Bernardini et al. [2] simulated RIT past a square roughness element also using DNS method, and found that the RIT process was controlled by a Reynolds number based on the momentum deficit
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past the obstacle, which accounts for the effect of roughness shape. In a previous study conducted in our laboratory
[11]
, we found that for an array of ramp elements, the horseshoe
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vortices detached from two ramp elements interact with each other and grow stronger due to counter-rotating effect. This suggests that RIT is enhanced compared to the case of an isolated ramp. In the present study, the effects on RIT
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mechanism in the hypersonic boundary layer according to three isolated roughness elements (ramp, diamond, and cylinder) were explored through DNS methodology based on a finite volume formulation. The drag increment and
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aero-heating rate on the roughness elements and the flat plane were also quantitatively measured and compared. The computational conditions were chosen based on measurements presented by Tirtey et al.
[8]
, with Reynolds number
of 2.6107/m.
The remainder of this paper is organized as follow. Section 2 discusses the numerical strategy, including the governing equations, high-order reconstruction, and code validation. The numerical results for RIT past the three
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isolated roughness elements discussed above are compared and analyzed in Section 3. The final section provides a brief summary and conclusion.
2. Numerical Strategy
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Our in-house code, UNITs (Unsteady NavIer-STokes equation solver), based on a cell-centered finite volume formulation with multi-block structured grids, is used to accurately resolve the unsteady flow. The convective terms are solved using the rotated Roe Riemann solver, through fourth-order MDCD-WENO (minimum dispersion and controllable dissipation-weighted essential non-oscillatory) reconstruction for the characteristic variables. A
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modified, fully implicit, lower-upper symmetric Gauss-Seidel (LU-SGS) with Newton-like sub-iteration in pseudo time is selected as the time marching method when solving the Navier-Stokes (N-S) equation.
2.1 Governing Equations and Numerical Method
The 3-D compressible N-S equation can be written as follows:
(1)
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U Fi Fvi t xi xi
where U, Fi, and Fvi represent the flow variable vector, inviscid flux vector, and viscous flux vector (i=1, 2 3), respectively. They are given as follows:
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ui 0 u u u p 1,i 1 i 1i 1 U u2 ，Fi u2ui p 2,i ，Fvi 2i u3ui p 3,i 3i u3 u q E i ( E p)ui ji j
(2)
where is the fluid density, ui (i=1, 2, 3) are the three velocity components, and variables p and E represent the fluid pressure and total energy. E p / [ ( 1)] ui ui / 2 , where is the specific ratio (1.4, in our case.) For the perfect gas, p=RT, where T is static temperature and R is a constant of 287.
ij is the stress tensor, expressed as follows:
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ui u j x j xi
ij
u k ij xk
(3)
where the molecular viscosity coefficient is determined by Sutherland’s law. is the second viscosity coefficient, typically assumed to be -2/3.
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The heat conduction term is determined by Fourier’s law qi kh T xi , where kh is c p / Pr . The Prandtl number Pr is set to 0.72.
2.2 High-order MDCD Reconstruction
The inviscid fluxes are solved using the rotated Roe Riemann solver combined with fourth-order MDCD-WENO
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reconstruction. MDCD is a linear reconstruction technology developed by Ren et al.
[12], [13]
that can achieve
minimum dispersion and controllable dissipation. In order to ensure shock-capturing properties, it is applied as a linear part of WENO schemes with symmetrical stencils. Details regarding the MDCD-WENO scheme can be found
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in References [12, [13].
2.3 Code Validation
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Two of our previous studies validated the DNS with MDCD-WENO scheme that we utilized in this study [11], [14]. Here, we will discuss some comparisons between our study and an experiment on the supersonic RIT past a
Disturbance Tunnel.
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diamond-shape roughness element, conducted by Kegerise et al.[15] in the NASA Langley Supersonic Low
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The Reynolds number (Re) is 10.8 106/m and Mach number (Ma) is 3.5. The free stream and the wall temperature are 92.55 and 290 K, respectively. The roughness height is about 0.48 times of the local undisturbed
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laminar boundary layer thickness, and it is located 41.2mm downstream of the leading edge of the flat plate. The total computational grids are about 67.2 million, y+wall is around 0.15~0.3, and x+ and z+ are about 2~5. A series of slow acoustic waves with random magnitudes, ranging from 1.0 kHz to 2500 kHz in frequency space,
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The flow structures around and behind the roughness element can be visualized utilizing the Q criterion, which is defined as the second invariant of the velocity gradient tensor [16]: Q
u v u w v w u v u w v w x y x z y z y x z x z y
(4)
As shown in Fig. 1, the characteristic flow structures around isolated roughness elements are the vortex tubes in
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the near wake, and the hair-pin vortices in the downstream region[17] of x>110mm. Between 110mm < x < 135mm, regular hair-pin vortices are generated by base flow deformation due to the instability nonlinear development. In this simulation, the disturbance saturation exhibits at about x=135mm, at which point the regular hair-pin vorticity near
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the symmetrical plane begin to split into more vortices.
Fig. 1: Vortex structures of Ma3.5 diamond element by iso-surface of Q-criterion
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The comparisons of mean mass flux (m/me) and R.M.S. mass flux (m’/mmax) of 104 kHz at x=110.6mm are
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presented in Fig. 2. As expected, a low velocity streak near the symmetrical plane due to the roughness blockage can be observed. At the same time, the high-velocity streaks approach to the wall due to high-speed flow downwash. A series of instantaneous flows at x=110.6 mm were recorded, and the most amplified disturbance (104 kHz) was
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identified through Fast Fourier Transformation(FFT) analysis. The DNS results match the measurements very well. The mass-flux (normalized by corresponding boundary layer edge value) profiles on the symmetric plane after
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the roughness are shown in Fig. 3. The numerical results show reasonable agreement with experiment until x=110.6mm, where the instability growth is nearly linear and flow is laminar. After x= 136.5mm, the disturbances
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saturate and transition happens, the numerical results show a bit of discrepancy with experiment.
Fig. 2: Comparisons of mean mass flux (m/me) and R.M.S. mass flux (m’/mmax) of 104 kHz at x=110.6mm 7
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Fig. 3 The normalized mass-flux (m/me) profiles on symmetric plane after the roughness element
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Fig. 4: Streamwise evolution of peak r.m.s. mass-flux fluctuations on symmetric plane
Figure 4 shows the streamwise evolution of peak r.m.s. mass-flux fluctuations on symmetric plane in the
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diamond roughness element’s wake. The dashed line in the figure denotes the roughness location, the symbols denote the measured data for the most amplified frequency of 100 kHz, and the solid line denotes the simulated peak mass-flux fluctuation of 104 kHz. The disturbance experiences linear growth before x=110.6mm, then the growth rate slows down due to nonlinear effect. The simulation gives a close enough disturbance saturation position with experiment, located at x136.5mm. After saturation, the disturbance begins to decay through breakdown. The very close agreement between experimental data and our simulation confirms that our DNS code is capable of simulating the RIT problem accurately, and the grid scale is small enough for current computation. The slight discrepancy in 8
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the highest amplified frequency may have been caused by the differences of inflow disturbance frequency characteristic between simulation and experiment.
3. Results and Discussions
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The computational setups we established in this study are presented below, followed by a discussion on the grid density and computational domains that were used in our computations. The numerical flow-fields around the ramp are then compared with measurements available in the literature. The flow patterns around three roughness elements, the effects of roughness shape on transition mechanism, aerodynamic and aero-thermodynamic performance are analyzed and compared.
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3.1 Computational Setup
Fig. 5: Sketch of flat plate and roughness elements
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Table 1: Geometrical parameters of ramp and flat plate
Ld
Wp
h
d
60mm
100mm
34.1mm
0.8mm
3.2mm
100
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Lu
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The RIT by three isolated roughness elements (ramp, diamond, and cylinder) mounted on a flat plate was investigated in this study; our process was very similar to experiments conducted by Tirtey et al.
[8]
. The plate used
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for most of the simulations is 60mm long and 34.1mm wide, and the plate used for experiments is 260mm long and 100mm wide. The distance between the roughness and the plate leading edge (Lu) is the same in the simulation and experiment (60mm). The height (h) of the roughness elements is 0.8mm, which is almost equal to the local undisturbed boundary layer thickness without the roughness. For the ramp, the width of the leading edge (d) is 3.2mm, which is four times that of the trailing edge; the ramp angle is 10 degrees. The widths of the diamond and cylinder elements (d) are both 3.2mm. Note that in order to be appropriately comparable with the other two elements, the cylinder roughness in simulation is a little narrower than the experimental cylinder (4mm). The geometric 9
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parameters of these three elements are presented in Fig. 5 and Table 1. The filled circles on the roughness elements represent the original points in the numerical simulations. Three unite Reynolds numbers were adopted in Tirtey’s experiments
[8]
: 9.2106, 1.8107, and 2.6107/m. In
this study, we only tested the largest Re of these. The flow parameters, including the Mach number, density, pressure, temperature, boundary layer thickness, unit Reynolds number, and Reynolds number based on the height
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(Reh =uh /) are listed in Table 2. The quiet inflow boundary condition was adopted for all the cases in our simulations, so the boundary layer transition was merely induced by the instability created by roughness elements themselves.
The unit length was 1 meter and the freestream velocity (U) was taken as the referenced velocity. The non-
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dimensional time step was taken as 0.0001, corresponding to 9.937910-8 second. To achieve second order in time marching, eight sub-iterations were applied to ensure the inner residuals decrease 2 to 3 orders. Ten flow-through times, defined as the ratio of the streamwise length over the free-stream velocity, were taken to compute and average. It took approximately 50,000 CPU hours to finish one simulation based on the Fine mesh. Table 2: Flow parameters
p(Pa)
0.0122
2449.42
T(K)
Tw(K)
(mm)
Re (m-1)
Reh
70
300
0.79
2.6107
1.6104
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6
(kg/m3)
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Ma
3.2 Effects of Grid Scale for Ramp Case
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In this subsection, three sets of grids, “Coarse”, “Medium”, and “Fine”, were adopted to compare the effects of
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grid density. For all the grids, the first grid point off the wall is about 10µm, corresponding to y+ 0.15~0.78 and about 100 grid points are distributed in the normal (y) direction with most of them in the boundary layer. In the downstream wake region, the grid scales for the Medium grid are 50µm in the streamwise (x) and
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spanwise (z) directions, corresponding to x+ and z+ 0.75~3.9. This grid density is comparable to the typical mesh adopted in similar DNS studies on RIT, such as those by Bartkowicz [18], Iyer et al.[19] and Bernardini et al.[20]. The primary difference between the Medium and the Coarse grids is the grid scale in the core flow region, where the latter is two times that of the former in streamwise and spanwise directions. In fact, the grid scales of the Fine and Medium grid are almost the same. The core flow region of Fine grid is about 4 times wider in the spanwise direction, in order to cover the wider wake of diamond and cylinder roughness. 10
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To save computational resource, the Fine grid only extends to 100mm downstream of the ramp, while Coarse and Medium grids extend to 230mm. The total grid numbers of Coarse, Medium, and Fine are 20, 27, and 50.7 million,
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respectively. The typical grids around the ramp are presented in Fig. 6 and grid parameters are listed in Table 3.
Fig. 6: Near-field grids around ramp roughness element
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Table 3: Grid parameters in wake core region for different grids
x+, z+
Downstream length (mm)
Total grid number (million)
Coarse
0.15~ 0.78
1.5~7.8
230
20
Medium
0.15~ 0.78
0.75~3.9
230
27
Fine
0.15~ 0.78
0.75~3.9
100
50.7
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y+wall
Figure 7 shows the skin friction coefficients (Cf) in the wake based on the three grids, which are nearly the same
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before x=10mm where the flow is still laminar. Between 10 and 100mm, in which region flow transition occurs, the Cf coefficients gradually increases to peak value, then decreases to a theoretical turbulent Cf value on a flat plate. The differences in Cf indicate that the flow transition occurs slightly downstream, about 10 mm, when simulating
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with the Coarse grid. The Cf coefficients by Medium and Fine grids are practically identical. In the further
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downstream region, Cf coefficients by Coarse, Medium, and Fine grids approach the theoretical turbulent skin friction value on the flat plate. Comparisons of five streamwise velocity profiles at x=10, 20, 30, 40, 50mm on the symmetric plane (z=0) and
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on the plane of z= 0.4mm are presented in Fig. 8. In the linear and weakly nonlinear disturbance growth region, where x is less than 40mm, the differences among the three grids are very small. At x=50mm, the nonlinear disturbance growth is dominated and the difference between the Coarse and Medium grids is more prominent than that between Medium and Fine grids. It indicates that the grid convergence is achieved and the scale of Fine grid is sufficient to resolve the transitional flow in the wake of roughness elements.
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Recall that the Fine grid density is better than that of some typical DNS simulation
[18][19][20]
. In this study, the
transition region and tendency are almost the same for the three grids, except that the transition location is a little downstream for the Coarse grid. For this reason, we assert that Medium and Fine grid scales are small enough to resolve the RIT problem. The following discussions on flow features and the transition mechanism are based on the
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grid scale of the Fine grid.
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Fig. 7: Cf distributions in wake region on the symmetric plane
Fig. 8 Streamwise velocity profiles on the symmetric plane (top row) and z=0.4mm plane (bottom row)
3.3 Qualitative Comparison with Experiment for Ramp Case
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In the experiment
[8]
, infrared thermography measurement, oil, and Acenaphthene sublimation visualization
techniques were adopted to depict the transition procedure after the roughness element. Infrared thermography and sublimation techniques can be used to highlight the high heat flux region, and oil visualization applied to depict the high skin friction region. As we know, the convective heat-flux Qw to the wall is significantly higher in turbulent boundary layer than
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laminar boundary layer, so Qw is typically used to judge the laminar or turbulent state of the flow. The nondimensional heat-flux rate is the Stanton number. We adopted the modified Stanton number for convenience, where freestream density and velocity were used instead of the boundary layer edge conditions, and total temperature was used to replace adiabatic wall temperature. The modified Stanton number is defined as follows:
Qw u cp (T0 Tw )
(5)
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St
For the ramp case, comparisons between Stanton numbers and skin friction coefficients in the wake for computations and measurements are shown in Fig. 9.
The computational flow topological structure is the same as that of the experiment in the vicinity of the
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roughness element. Two high heat-flux rate and skin friction streaks exist due to the vortex tube, then the two streaks combine into a single steak between x=40 and 80mm in the simulation, where the flow transition begins.
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Near the outflow, the two streaks spread out in the spanwise direction. It seems that the vortex tube near the roughness is a little shorter in the experiment, and near the outflow, the
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disturbed flow region is a little wider than the computational result. It should be noted that the experiment was conducted in a noisy wind tunnel, but the quiet inflow condition was applied in the DNS study. Differences in
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freestream conditions may have caused some deviation in flow topology.
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Fig. 9: Comparisons of Stanton number St and skin friction coefficient Cf distributions (Top: Simulation; Bottom: Measurements by infrared thermography and oil-visualization)
3.4 Mean and Instantaneous Flow-fields Around the Three Roughness Elements Because diamond and cylinder geometries differ from that of the ramp, the grid distributions around them
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also differ, as shown in Fig. 10. The grid scales in the wake are the same as the Fine grid for the ramp.
Fig. 10: Grids around isolated diamond and cylinder roughness elements
Figures 11 and 12 present the mean flow patterns in the near-field of roughness elements. Any roughness element in the boundary layer will generate an adverse pressure gradient, which may cause separation before it. For
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the ramp, a single upstream separation bubble is observed. The adverse pressure gradient before the diamond is a little stronger, and two pairs of separation bubbles are observed. As the bluntness of the cylinder is the largest
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among three elements, its resultant adverse pressure gradient is the largest, generating three pairs of unstable separation bubbles. The flow diverges very rapidly after the roughness element, forming the low-pressure region,
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where a main recirculation can be easily observed. In hypersonic or supersonic boundary layers, a shock wave forms before the roughness element due to
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supersonic/hypersonic flow. For the cylinder, an obvious separation shock forms before the main separation bubble
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near the roughness element, similar to cases with higher cylinders [5][14][18].
Fig. 11: Mean streamlines and pressure distribution on symmetric plane (from left to right: ramp, diamond, cylinder)
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According to the comparisons of surface streamlines, the horse-shoe vortices past the square and cylinder are much stronger and wider than that of ramp. At the same time, the Stanton numbers at the nose region are also much
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larger than that of ramp.
Fig. 12 Surface streamlines and contours of Stanton number around the three roughness elements
As shown in Fig. 11, a strong “bow shock” is observed before the roughness element in the hypersonic boundary
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layer. At the same time, a series of separation shocks presents due to the separation bubbles. Expansion waves and reattachment shock waves are formed after the roughness element.
Shocks should be plotted clearly, despite the fact that they are very complex and difficult to fully demonstrate. Fortunately, they can be fairly readily illustrated using the “normal Mach number” parameter (Man)
[21]
. The iso-
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surface of Man=1 stands for the shock wave, defined as follows:
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(6)
Fig. 13: Shock wave around roughness elements
As shown in Fig. 13, for the ramp, the upstream separation bubble is very small and the separation shock is
restricted to a narrow region along the spanwise direction. Both the separation and reattachment shocks are convex. For the diamond, the separation shock is the widest due to the largest upstream separation flow (Fig. 12 and Fig. 14) and the downstream reattach shock exhibits a concave shape. The shock system for the cylinder is similar with those
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past the diamond, apart from the flat reattachment shock. The separation shock and bow shock join to form a single shock surface at 1.53, 3.17 and 3.83 mm downstream, for ramp, diamond and cylinder, respectively. The separation bubbles around these three roughness elements are illustrated below using the iso-surface of a small negative streamwise velocity (-110-4 U), colored by the streamwise Reynolds stress u’u’ , as shown inFig. 14. For the ramp, the upstream separation flow is restricted to a relatively small zone about 1.98mm long and
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2.32mm wide. The small u’u’ on the separation flow surface indicates that the separation may be quite steady. After the ramp, the separation bubble is about 3.72mm long, and larger u’u’ is observed.
For the diamond, the separation extends about 3.91 upstream and 11.22mm in the spanwise direction. The u’u’ is small near the symmetric plane, but gradually increases in magnitude as moving away from the symmetric plane.
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The downstream separation is about 7.6mm long, and its u’u’ is at a higher level than that at the upstream separation flow. For the cylinder, the upstream separation is the longest (about 5.25mm) among the three roughness elements, and 6.85mm wide. The u’u’ before the cylinder is about two orders higher than that past the diamond, and the
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downstream recirculation is the shortest, about 3.02mm in the streamwise direction.
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Fig. 14: Mean separation bubbles around three roughness elements
Fig. 15: Mean vortices near roughness elements colored by streamwise velocity and coefficients of skin friction
The vortices in space around the roughness elements are shown in Fig. 15. With increasing bluntness of the roughness element, the adverse pressure gradient formed on the windward side of the obstacle also increases, 16
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pushing the flow farther away from the elements. The wakes of the diamond and cylinder are wider than that of the ramp due to the stronger horseshoe vortices; the horseshoe vortices of the diamond reach a farthest location in the
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spanwise direction.
Fig. 16: Instantaneous non-dimensional temperature at sections z/d=0 and y/h=0.5
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(from top to bottom: ramp, diamond and cylinder, respectively)
Flow structures at the spanwise (z/d=0) and normal (y/h=0.5) sections are shown by temperature contours in Fig.
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16. The roughness element in the boundary layer slows down the main flow and several low-velocity and high temperature streaks are then observed behind the roughness. The high-velocity flow out of the boundary layer and
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the low-velocity flow in the wake form a very strong 3-D shear layer near the symmetric plane, which is very stable near the roughness before x=35mm. In the regions further downstream (from 35 to 60mm) some regular structures
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arise, indicating the nonlinear instability breakdown. In the region approaching the outlet (about from 60 to 90mm) the regular structures gradually develop, evolve, and split into many small-scale and chaotic structures. The similar
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phenomena were likewise observed in studies on cylindrical [5][18] and hemispherical roughness elements [19]. In the y/h=0.5 section, the most distinct structures are the horseshoe and wake vortices. For the ramp, almost no
horseshoe vortices are observed; for the diamond, the horseshoe vortices are very stable. For the cylinder, these horseshoe vortices are unstable due to strong disturbance from the cylinder, and nonlinear instability breakdown starts at about x=20mm (about 15 mm farther upstream than those of the shear layer on the symmetric plane.) At the
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end of the computation domain (100mm,) the width of the disturbed wake for the cylinder is the widest, about 16mm; for the ramp and diamond, they are about 8 and 12mm, respectively. As shown in Fig. 17, the most distinct structures are the stable wake vortex pairs from both sides of the roughness just beside the symmetric plane. For diamond and cylinder roughness elements, horseshoe vortices are especially readily observed. The wake vortices grow weaker as moving downstream until break down into many
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nearly regular, coherent hairpin vortices, marking where transition occurs. Between the hairpin vortices and the flat plate, a few small-scale streamwise vortices can also be found. The horseshoe vortices of the diamond are farther from the mid-plane than those of the cylinder. And for the ramp, the horseshoe vortices are much closer to the
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roughness and more difficult to distinguish.
Fig. 17: Instantaneous vortex structures using iso-surface of Q criterion colored by streamwise velocity (from top to bottom: ramp, diamond, cylinder)
Unsteady pressure fluctuations on the flat plate in the separation zone before and behind the roughness were
recorded as shown in Fig. 18. For the ramp, the sample point before the roughness was located at (-4.5 0, 0mm) and the sample point after the roughness was located at (2.3, 0, 0mm). For the diamond and cylinder, the upstream and downstream sample points were the same, (-2.4, 0, 0mm) and (3.4, 0, 0mm), respectively. According to comparative 18
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analysis of power spectral density (PSD), the upstream and downstream separation bubbles show some distinct periodic motions; the dominating frequencies are about 55.27, 30.19, and 42.58 kHz for the ramp, diamond, and cylinder, respectively. For the ramp, the unsteady flow exhibits several notable harmonic frequencies, including those at 110.54, 165.82, and 221.09 kHz. For the diamond and cylinder, however, there are no such harmonic
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frequencies. The magnitude of fluctuation energy for the ramp is the smallest, while that of cylinder is the largest.
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Fig. 18: Normalized pressure fluctuation before and after roughness elements
3.5 Effects on Transition
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Figure 19 shows the streamwise Reynolds stress (u’u’) distribution at four streamwise sections (x=10, 40, 70, and 100 mm, respectively) in the roughness wake. The contour lines of us
u
y u z , which reflect the 2
2
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local shear level, are also plotted in the same figure. By comparing information shown in Fig. 15 and Fig. 19, we found that us distributes around the low-velocity
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streak forming 3-D high shear level vortices. These 3-D shear layers show unstable characteristics, and the Reynolds stress distributes right around their boundaries.
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For the ramp, the disturbed region is narrow at x=10mm section. At x=40mm section, the instability of the wake
vortices grows larger and the horseshoe vortices near the flat plane become less stable. At x=70mm section, the horseshoe vortices are less stable than the wake vortices, and the instability zone becomes wider. At the outlet of the computational domain (x=100mm), the half width of the fully turbulent boundary layer is about 1mm. When z is greater than 1mm, the flow remains in the transitional state.
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For the diamond and cylinder, the horseshoe vortices are less stable than the wake vortices at x=10mm section, and the flow is constrained in the low-speed streak region. For the diamond, the instability of the wake vortices grows faster than that of the horseshoe vortices; for the cylinder, both the wake vortices and the horseshoe vortices play important roles in the flow transition. At the end of the computation domain (x=100mm), the half width of the
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fully turbulent wake is about 2.5mm for the diamond and about 6mm for the cylinder.
Fig. 19: Contours of streamwise Reynolds stress and line of us at four streamwise sections
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(from left to right: ramp, diamond, cylinder)
Fig. 20: TKE Contours in wake of roughness at y=0.4mm and Prms on flat plate (from top to bottom: ramp, diamond, cylinder) 20
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The lateral spread of the turbulent wedge is shown in Fig. 20, through TKE (turbulent kinetic energy) distribution on the y/h =0.5 (y=0.4mm) plane and the Prms (root mean square of pressure) distribution on the flat plate. The shapes of the transitional/turbulent wake are similar among all three isolated roughness elements. For the ramp, the transition begins at about 40mm near the symmetric plane, and for the diamond, the transition starts a little further downstream, (about 50mm,) also near the symmetric plane.
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For the cylinder, the flow transition is much more complex than those of the ramp and diamond. Near the symmetric plane, the transition occurs at about x=25mm. However, the flow reaches the transitional/turbulent state immediately after the roughness, due to the unsteady horseshoe vortices.
At the outlet of the computational domain (x=100mm), the half width of the transitional/turbulent wedge of the
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ramp is about 4mm, and about 6mm and 8mm for the diamond and cylinder, respectively.
Fig. 21: Streamwise TKE evolution
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(Left: Line integration on the symmetric plane; Right: Surface integration on the y-z plane)
The streamwise TKE evolutions are presented in Fig. 21. Two kinds of integration are applied: one is the line
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integration in the boundary layer, defined by TKELine = ò
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normal and spanwise directions, defined by TKESurface = ò
y=d y=0
ui¢ui¢ 2 dy , another is the surface integration in both
z=zmax z=zmin
ò
y=d y=0
ui¢ui¢ 2 dydz . It should be noted that the vertical
axis is in logarithmic form.
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The TKELine mainly represents the disturbance evolution on the symmetric plane, transported by the wake
vortices. A significantly linear growth rate is observed from 15 to 55mm for cylinder, from 15 to 40mm for ramp, and from 30 to 60mm for diamond, respectively. From 20 to 40mm, the TKEline of cylinder roughness is the largest, while its growth rate is the smallest. The growth rate of ramp is the largest with medium TKEline. The TKEline of diamond is the smallest with medium growth rate. The TKEline growth of ramp slows down at about x=40mm, but it happens at about x=60mm for both the diamond and cylinder. The saturation is achieved at about x=80mm for the 21
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ramp, where transition is finished. For cylinder and square, saturations are achieved at about x=70mm. When x is greater than 80mm, the TKEs of ramp, diamond, and cylinder are nearly the same. It indicates that the flow is fully turbulent. The TKESurface is the total TKE on the y-z plane, transported by wake and horseshoe vortices simultaneously. The curves exhibit the same disturbance linear growth region and tendency with TKEline results. The TKESurface of
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cylinder is the largest in the whole computational domain. It indicates that the disturbances by horseshoe vortices play a key role. The TKESurface of ramp is smaller than that of diamond before x=20mm, while it grows with the highest linear growth rate after x=20mm.
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3.6 Effects on Aerothermodynamics and Aerodynamic Performance
An ideal roughness element will trip the flow from laminar to turbulence very rapidly without producing much drag at the roughness itself or encountering severe aerothermal load (high temperature or aero-heating rate). We investigated not only the transition effects of the three roughness elements in our numerical simulation, but also the
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aerothermal load and drag increment caused by the roughness element.
Fig. 22: Stanton number on wall (from top to bottom: ramp, diamond and cylinder)
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Figure 22 presents the non-dimensional heat flux (Stanton number) distribution on the roughness and flat plate, where the computational flow topological structures are same as that of the experiment[8]. In the vicinity of the roughness elements, two high heat-flux rate streaks exist due to the steady vortex tube. For the ramp, the two streaks combine into a single steak between 40 and 80mm, where the flow transition begins. Near the outflow, the heat flux streak spreads in the spanwise direction. For the diamond, the two heat flux streaks near the symmetric plane
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separate, and the convective heat is higher than that of the ramp. Two high heat flux streaks are formed at the side region (z5.5mm) due to the horseshoe vortices, identical to the instantaneous temperature distribution shown in Fig. 16. For the cylinder, three pairs of separation bubbles form and wrap around the element. As a result, six clearly distinguishable high heat flux streaks are formed in the wake. The distributions of the heat flux are almost uniform
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after x80mm in the spanwise direction for all the three roughness elements.
The widths of the transitional/turbulent wake are about 8, 12, and 16mm for the ramp, diamond, and cylinder, respectively, coincident with the temperature and TKE distributions shown in Fig. 16 and Fig. 20. The temperature and pressure dramatically increase after the strong bow shock near the head of each roughness element. As a result, the roughness itself will encounter a severe aerothermal load. As shown in Fig. 12 and Fig. 22,
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the heat flux around the top of the roughness is massive compared to the nearby laminar zone. The value and
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location of the maximum heat flux on the roughness element surfaces are listed in Table 4. For the ramp, maximum heat flux at the top surface is 16.691 times of local undisturbed laminar heat value. For the diamond and cylinder,
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the maximum heat flux magnitudes are 133.691 and 173.793 times of laminar heat value，located at their windward vertices.
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Table 4: Maximum non-dimensional heat flux (103) for roughness elements Hmax
Hlam
Hmax/Hlam
Position
Ramp
4.106
0.246
16.691
Top
Diamond
32.888
0.246
133.691
Windward vertex
Cylinder
42.753
0.246
173.793
Windward vertex
Comparisons among the normalized heat flux at three spanwise sections are presented in Fig. 23. The solid, dashed, and dash-dot-dotted lines represent the ramp, diamond, and cylinder, respectively. On the symmetric plane 23
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(z/d=0), the heat flux of the ramp wake is larger than that of the diamond or cylinder from 10 to 80mm, reaching a peak (St1.510-3) at 65mm. When x is larger than 80mm, the heat fluxes of the three roughness elements reach a plateau around St=0.810-3, about 5~6 times that of the local laminar value, representing the typical level of turbulent flow. At sections of z/d=0.5 and 1.0, the heat flux of the ramp wake is the smallest due to its narrow
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transitional/turbulent wake, while the high heat flux streak of the cylinder is the widest.
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Fig. 23: Normalized heat flux along streamwise direction on three spanwise positions Table 5: Drag coefficients (104) for roughness elements Cd (t)
Cd (r)
Cd (f)
Cd_p (r)
Cd_v (r)
Ramp
8.449
0.124
8.325
0.104
0.020
Diamond
9.280
0.640
8.640
0.552
0.089
Cylinder
9.863
0.595
9.267
0.489
0.100
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The drag caused by the roughness was also evaluated as listed in Table 5. The referenced area is the area of flat plate (5.45610-3 m2). Cd (t), Cd (r), and Cd (f) represent the total drag, the drag by the roughness itself, and the drag of the flat plate, respectively, and Cd_p and Cd_v stand for pressure and viscous drag. The drag force by the roughness elements themselves are rather small compared to the overall drag, about 1.47,
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6.90, and 6.03% of the total drag for the ramp, diamond, and cylinder, respectively. Most of the total drag is comprised of the viscous drag on the flat plate, which is affected by the transitional/turbulent wake. For the roughness elements themselves, the drag force by diamond and cylinder are similar and much larger (about 5 times)
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than that provided by the ramp.
4. Conclusions
The hypersonic boundary layer transition induced by three types of isolated roughness element (ramp, diamond, and cylinder) has been explored in this study using the DNS method. The width and height of the three roughness
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elements are the same, and their heights approximately equal to the local undisturbed boundary layer. The flow transition is dominated by the instability of the counter-rotating streamwise vortices generated by the
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roughness. For the cylinder, the horseshoe vortices play a more important role than the wake vortices just behind the roughness. The transition distance downstream of the ramp is about 80mm, a little longer than that of the diamond or
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cylinder (70 mm,) and the width of the transitional/turbulent wake of the ramp is smaller than those of the other two obstacles, but the growth rate of the disturbance for the ramp is the largest. Conversely, the maximum heat flux and
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drag increment caused by the ramp are the smallest among the three roughness elements, while they are the largest for the cylinder.
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An ideal roughness element should trip the flow from laminar to turbulence very rapidly, without producing
much drag at the roughness itself or encountering severe aerothermal load. We posit that the ramp is a suitable roughness element for triggering hypersonic boundary layer transitions when accounting for transition effect, aerothermal load, and the drag increment among the three roughness elements we investigated.
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Acknowledgments This work was supported by the National Science Foundation of China (Grant number 11372159) and China Postdoctoral Science Foundation (Grant number 2015M571029). The authors thank Tsinghua National Laboratory
References
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for Information Science and Technology for the computational resources.
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[2] Bernardini M., Pirozzoli S. and Orlandi Paolo., “Parameterization of Boundary Layer Transition Induced by Isolated Roughness Elements,” AIAA Journal, Vol. 52, NO. 10, October 2014, pp. 2261-2269;
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