ABSTRACT. model for transitional/turbulent flows (AIAA Journal, Vol. 39, No. 9) for use in

Transcription

1 ABSTRACT MALECHUK, ANDREW MARTIN. Simulation of Transitional Flow over an Elliptic Cone at Mach 8 using a One-Equation Transition/Turbulence Model. (Under the direction of Jack R. Edwards and Hassan A. Hassan.) The purpose of this research has been to extend a previously developed oneequation model for transitional/turbulent flows (AIAA Journal, Vol. 39, No. 9) for use in the simulation of transitional/turbulent flows over three-dimensional bodies in conventional hypersonic tunnels. This is done computationally through the combination of the Spalart-Allmaras one-equation turbulence model and an eddy viscosity-transport equation based on that proposed by Xiao, Edwards, and Hassan for high disturbance environment (HIDE) induced transition. The blending of these two pieces of the model is achieved through the use of an intermittency function based on the work of Dhawan and Narasimha. The test case used in this research is an elliptic cone of aspect ratio : in a Mach 8 environment with Reynolds numbers between the range of.98x0 6 /ft and 6.09x0 5 /ft. Two separate methods are used to find the boundary layer edge flow properties under the resulting conical shock. The first of these methods uses fluid values extracted from the surface of the cone after an inviscid calculation. The second searches for the boundary layer edge by locating the largest momentum flux under the shock. The second of the two approaches is found to be the most successful in replicating transitional flow heat flux data measured experimentally by Kimmel, Poggie, and Schwoerk. Over the range of Reynolds numbers examined, the model reasonably predicts the location and extent of the transitional region, but does not effectively predict fluid properties within the transitional region.

2 SIMULATION OF TRANSITIONAL FLOW OVER AN ELLIPTIC CONE AT MACH 8 USING A ONE-EQUATION TRANSITION/TURBULENCE MODEL by ANDREW MARTIN MALECHUK A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Master of Science AEROSPACE ENGINEERING Raleigh 00 APPROVED BY: Dr. Jack R. Edwards Dr. Hassan A. Hassan D. Kailash C. Misra

3 BIOGRAPHY ANDREW MARTIN MALECHUK 75 N Fawn Forest LN Campus Box 790 Pittsboro, NC 73 NCSU (99) Raleigh, NC 7695 ammalech@eos.ncsu.edu (99) EDUCATION Masters Aerospace Engineering. (utational Fluid Dynamics Concentration). Minor Mathematics. North Carolina State University, Raleigh NC. GPA Thesis: Simulation of Transitional Flow over an Elliptic Cone at Mach 8 using a One-Equation Transition/Turbulence Model. B.S. Aerospace Engineering. North Carolina State University, Raleigh NC. Overall GPA - 3.6, Major GPA Graduation Date - May 000. RELEVANT COURSES utational Fluid Dynamics. Aircraft Design. utational Reactive/-Phase Flows. Incompressible Aerodynamics. Advanced ressible Aerodynamics. Boundary Layer Theory. Numerical Analysis. Incompressible and ressible Flow Labs. IMPORTANT PAPERS Sandia National Laboratories Summer 00 Internship Implementation of the NCSU Transition Model into the SACCARA ressible Fluid Mechanics Code. Cargo Lifter, Inc. Summer 000 Internship Preventing Snow and Ice Accumulation on Airships. ii

4 BIOGRAPHY (continued) WORK EXPERIENCE 8/00-/0 NC State University Research Assistant - Raleigh, NC Modified existing parallel computational fluid dynamics code to include NCSU one-equation transition/turbulence model. Examined effects of various parameters on location and extent of flow transition using hypersonic 3-D cone and modified code. Lectured Aerospace Engineering classes. 6/0-8/0 Sandia National Laboratories Intern - Albuquerque, NM 5/0-8/0 Added NCSU transition/turbulence model to SACCARA CFD code. Obtained L Security Clearance while working in department 95. Presented internship work at SNL symposium receiving 94 out of 00 possible grade points. 6/99-/00 CargoLifter Inc. Intern - Raleigh, NC Researched effects of inclement weather on airships. Created airship database/library. Performed market studies. SOCIAL Activities Senior Aircraft Design Team Leader. Supervisor of annual special needs children s beach retreat. Ground crew for annual Winston-Salem, NC air show. Encounter College Retreat leader and coordinator. NC State Intramurals. Awards 8 Semesters on Dean s List. Sigma Gamma Tau Honor Society Member. COMPUTER EXPERIENCE Systems UNIX, LINUX. Software Microsoft Office, ANSYS, Tecplot, Word Perfect, MATLAB, MAPLE, Pro Engineer, AutoCAD. Programming Languages FORTRAN (MPI). iii

5 ACKNOWLEDGEMENTS Andrew Malechuk would like to acknowledge the support of North Carolina State University, specifically the guidance provided by Dr. Jack R. Edwards and Dr. Hassan A. Hassan. The author would also like to express appreciation to Dr. Christopher Roy and the rest of the technical staff of Department 95 at Sandia National Laboratories for many helpful discussions. This research is sponsored by Sandia National Laboratories under Contract BF-856. IBM SP- computer time is provided by a grant from the North Carolina Supercomputing Center. iv

6 TABLE OF CONTENTS LIST OF TABLES... vi LIST OF FIGURES... vii NOMENCLATURE... ix INTRODUCTION.... Importance of Modeling Transition.... Previous Research....3 Objectives of Research... 4 MODEL DESCRIPTION Introduction Eddy Viscosity-Transport Equation for Non- Turbulent Fluctuations Spalart-Allmaras Turbulence Model Transition Region Modeling Transition Criteria....6 lete Transition/Turbulence Model....7 Calculation of Eddy Viscosity CALCULATION DETAILS Elliptic Cone Grid Determining Boundary Layer Edge Value Code Description and Boundary Conditions Initialization and Run Parameters RESULTS AND DISCUSSION erimental Data Calculation of Heat Flux Convergence Fully Laminar and Turbulent Flow Transitional Region arison of Methods... 5 CONCLUSIONS Conclusions... 5 LIST OF REFERENCES... 8 APPENDIX... 9 TABLES FIGURES... 3 v

7 LIST OF TABLES Table : Freestream properties for simulations Table : Transition location and magnitude with respect to experimental data vi

8 LIST OF FIGURES Figure : Elliptic cone grid orientation... 3 Figure : Velocity distribution, X=0.04m, θ = 6.7 o (Re/L=.98x0 6 /ft, transitional)... 3 Figure 3: Pressure distribution, X=0.04m, θ = 6.7 o (Re/L=.98x0 6 /ft, transitional)... 3 Figure 4: Momentum distribution, X=0.04m, θ = 6.7 o (Re/L=.98x0 6 /ft, transitional)... 3 Figure 5: Convergence history (Re/L=.98x0 6 /ft, fully laminar and turbulent) Figure 6: Turbulent residual norm history (Re/L=.98x0 6 /ft, transitional) Figure 7: Turbulent residual norm history (Re/L=.03x0 6 /ft, transitional) Figure 8: Turbulent residual norm history (Re/L=7.8x0 5 /ft, transitional) Figure 9: Turbulent residual norm history (Re/L=6.09x0 5 /ft, transitional) Figure 0: Pressure distribution X=0.933m (Re/L=.98x0 6 /ft, fully laminar) Figure : Temperature distribution X=0.933m (Re/L=.98x0 6 /ft, fully laminar) Figure : Temperature distribution X=0.933m (Re/L=.98x0 6 /ft, fully turbulent) Figure 3: Heat flux distribution along rays of cone (Re/L=.98x0 6 /ft, fully laminar) Figure 4: Heat flux distribution along rays of cone (Re/L=.98x0 6 /ft, fully turbulent) Figure 5: Transition criteria over elliptic cone (Re/L=.98x0 6 /ft, Tu=.575) Figure 6: Intermittency function over elliptic cone (Re/L=.98x0 6 /ft, transition, Tu=.575) Figure 7: Viscosity distribution θ =63 o (Re/L=.98x0 6 /ft, transitional, Tu=.575) Figure 8: Non-turbulent time scale X=0.060m (Re/L=.98x0 6 /ft, transitional) Figure 9: Heat flux distribution along rays of cone (Re/L=.98x0 6 /ft, ISDE, Tu=.5) Figure 0: Heat flux distribution along rays of cone (Re/L=.98x0 6 /ft, BLES, Tu=.5) Figure : Heat flux distribution along rays of cone (Re/L=.98x0 6 /ft, BLES, Tu=.575)... 4 Figure : Heat flux distribution along ray of cone (Re/L=.98x0 6 /ft, θ =45 0, Tu=.575)... 4 vii

9 LIST OF FIGURES (continued) Figure 3: Heat flux distribution along rays of cone (Re/L=.03x0 6, ISDE, Tu=.5)... 4 Figure 4: Heat flux distribution along rays of cone (Re/L=.03x0 6 /ft, BLES, Tu=.5)... 4 Figure 5: Heat flux distribution along rays of cone (Re/L=.03x0 6 /ft, BLES, Tu=.575) Figure 6: Heat flux distribution along rays of cone (Re/L=7.8x0 5 /ft, ISDE, Tu=.5) Figure 7: Heat flux distribution along rays of cone (Re/L=7.8x0 5 /ft, BLES, Tu=.5) Figure 8: Heat flux distribution along rays of cone (Re/L=7.8x0 5 /ft, BLES, Tu=.575) Figure 9: Heat flux distribution along rays of cone (Re/L=6.09x0 5 /ft, ISDE, Tu=.5) Figure 30: Heat flux distribution along rays of cone (Re/L=6.09x0 5 /ft, BLES, Tu=.5) Figure 3: Heat flux distribution along rays of cone (Re/L=6.09x0 5 /ft, BLES, Tu=.575) viii

10 NOMENCLATURE a = non-turbulent fluctuation constant (first-mode) b = non-turbulent fluctuation constant (second mode) C b = Spalart-Allmaras model constant (0.355) C b = Spalart-Allmaras model constant (0.6) C p = specific heat coefficient C t = model constant (0.35) C t3 = Spalart-Allmaras model constant (.) C t4 = Spalart-Allmaras model constant (0.5) C w = Spalart-Allmaras model constant (3.39) C µ = model constant (0.09) c = model constant ( ) c = model constant c 3 = model constant (0.0000) c 4 = model constant (0.5) c 5 = model constant (0.000) d = distance to nearest wall f t = turbulence function in Spalart-Allmaras model f v = wall damping function in Spalart-Allmaras model f w = wall blockage function in Spalart-Allmaras model k = non-turbulent fluctuation energy n = normal vector Pr = Prandtl number (0.7) ix

11 NOMENCLATURE (continued) p = pressure q w = wall heat flux Re = Reynolds number R T = transition criteria function S = rotation tensor s = surface distance s T = surface distance to transition location Tu = average free-stream fluctuation intensity T = temperature t = time U = total velocity u = x-axis velocity δ * = displacement thickness Γ = intermittency function κ = von Karman constant (0.4) µ = viscosity ν = kinematic viscosity ν nt = non-turbulent eddy viscosity ν t = eddy viscosity v% = transported eddy viscosity in Spalart-Allmaras model Ω = rotation tensor magnitude ω = disturbance frequency x

12 NOMENCLATURE (continued) ρ = density σ = model constant ( ) σ l = diffusion term function based on flow regime σ t = diffusion term function based on flow regime τ = non-turbulent time scale Subscripts l = laminar = first-mode = second-mode e = boundary layer edge HIDE = high disturbance environment j = tensor notation index k = kinetic energy l = laminar max = maximum min = minimum nt = non-turbulent T = transitional t = turbulent w = wall = freestream xi

13 NOMENCLATURE (continued) Superscripts * = boundary layer reference state ~ = working variable = vector xii

14 INTRODUCTION. Importance of Modeling Transition Since the early days of aero-sciences it has been known that two regimes of fluid flow exist: laminar and turbulent. In many real world situations though, both conditions exist simultaneously, with a fluid flow transitioning from laminar to turbulent somewhere along the length of an object in the flow. The location of transition onset is extremely important in the design and analysis of hypersonic vehicles since many important parameters are dependent upon whether the conditions are transitional, laminar or turbulent. Specifically, experimental data has shown that skin friction and heat transfer tend to be larger in the turbulent region than in the laminar region. Peak values of skin friction and heat transfer may, however, occur within the transitional region. Therefore it is important not only to locate the points at which transition onset takes place but also to determine the spatial extent over which transition to turbulence occurs. With these facts in mind, the importance of correctly predicting the transition location and extent in a particular flowfield is apparent. Therefore, it should be no surprise that development of a computational fluid dynamics (CFD) model that accurately predicts transitional flows is held to be one of the most important goals of the CFD discipline.. Previous Research In recent years there has been a noticeable advancement in the development of transitional flow prediction models designed for use in CFD codes. North Carolina State University s Aerospace Engineering Department has been one of the leaders in the effort to develop and perfect a model that is usable in a wide variety of flowfields. One attempt at creating a transitional model for first-mode disturbances was performed by Warren and

15 Hassan []. This was done through the combination of a kinetic energy (k) and enstrophy (ζ) turbulence model with a model for the growth of laminar fluctuations. Using the minimum skin friction or heat transfer as the location of transition (due to the correlation of these properties and transition onset location as stated in Section.) was found to be limited to simple shapes and the absence of adverse pressure gradients. The research determined that the transition onset location along the test body is more accurately determined for a wider range of environments by determining the point at which the ratio of turbulent and non-turbulent kinematic viscosities as shown in Equation (.) first exceeds unity when moving downstream from the front of the geometry. v~ = (.) C v R T µ Dhawan and Narasimha s intermittency expression [] was used to mix the non-turbulent and turbulent model characteristics in the transitional region of the flow. This model was then extended to second-mode disturbances be McDaniel, Nance, and Hassan [3]. One weakness of the model is that the derived first-mode and second-mode timescales (Equations (.) and (.3)), which drive the growth of laminar fluctuations are based on a curve fit using the average turbulence intensity, Tu, of the flow: τ = a / ω a= ( Tu 0.38) (.) τ = (.3) b / ω This curve fit is only accurate for Tu less than or equal to 0.5, limiting the model s use to low disturbance-environment conditions. Using knowledge gained from the two-dimensional simulations, the k-ζ transition/turbulence model was advanced further by Xiao, Edwards, and Hassan [4] in an

16 attempt to define the disturbance mode that bypassed the first and second-modes and to extend the model to three-dimensions. The result was the definition of a high disturbance environment (HIDE) mode that was found to dominate the other modes of transitional flow for high Mach numbers and larger values of Tu. The average turbulence intensity is still used to define the laminar fluctuation time scale that drives the non-turbulent fluctuation model, but the time scale is also dependent upon the local Reynolds number and boundary layer characteristics: τ = c HIDE ν Re c Ue c * δ = Tu /8 (.4) The boundary layer edge values for the calculation of the Reynolds number and displacement thickness were found by extracting and saving surface flow parameters along the test subject from a converged inviscid run with the code. An elliptic cone with aspect ratio of : at Mach 8 with Tu=.5 was used as the test case with matching experimental data provided by Kimmel, Poggie, and Schwoerke [5]. A minimum heat transfer criteria was used around the circumference of the cone to define the transition onset location. Only one quarter of the cone flowfield was simulated because of the bilateral symmetry of the geometry. When compared with experimental heat transfer data for three rays along the cone (0 o, 45 o, and 88 o ) over a variety of Reynolds numbers, the modified model performed well at predicting not only the transition onset location, but also the spatial extent over which transition occurred. With the success of the k-ζ transition/turbulence model having been established, Edwards, Roy, Blottner, and Hassan [6] proceeded to develop a one-equation transition/turbulence model based on combining an eddy viscosity-transport model for 3

17 non-turbulent fluctuation growth with the Spalart-Allmaras turbulence model. Since the Spalart-Allmaras model is a single equation, is relatively insensitive to normal grid spacing at the wall, and produces a good mixture of accuracy and robustness for wallbounded turbulent flows, it is a very desirable model to use in a transition/turbulence model. Since most of the initial test cases involved turbulence intensities less than 0.5, a curve fit similar to that used in Warren and Hassan [] was once again used to define the laminar time scale: τ = a / ϖ a = ( Tu) ( Tu) Tu = 0.03 (.5) The criterion for transition in Equation (.) was used to determine the location of transition onset in the flowfield. The model was applied with reasonable success to transitional flows over a flat plate, a supercritical airfoil, a multi-element airfoil in landing configuration, a circular cylinder, and a sharp cone, all of which are twodimensional or axisymmetric flows. The predictions were found to be very similar to those obtained using the k-ζ transition/turbulence model. The success of this research produced the desire to examine the effectiveness of the Spalart-Allmaras transition/turbulence model in a hypersonic, three-dimensional, high turbulence intensity environment..3 Objectives of Research Past research has resulted in the development of reasonably accurate transition/turbulence models based on the k-ζ and Spalart-Allmaras turbulence models. The next step in this process, and the objective of the research discussed in this paper, is to develop an extension of the one-equation transition/turbulence model that is valid for 4

18 three-dimensional flows with high average turbulence intensities. This involves the modification of the non-turbulent component of the model to include a time scale characteristic of high disturbance environment fluctuation growth in a manner similar to that used in the three-dimensional version of the k-ζ transition/turbulence model. The test case for this research is again flow over an elliptic cone with aspect ratio of : at Mach 8 with Tu=.5. This case is useful in that it contains heat flux measurements across a wide range of Reynolds numbers [5]. arisons with this data will provide a first assessment of the effectiveness and accuracy of the extended one-equation model. MODEL DESCRIPTION. Introduction In an effort to preserve the structure of the earlier model, as much of the twodimensional Spalart-Allmaras transition/turbulence model as is possible is left unmodified and is transferred directly to the three-dimensional, high average turbulence intensity model. The major modifications involved in the conversion of the model involve the redefinition of the eddy viscosity-transport equation in the non-turbulent region of the flow to account for HIDE-driven transition. Since the three-dimensional HIDE version of the k-ζ transition/turbulence model [4] provides reasonably accurate results, its model for non-turbulent fluctuation growth is transformed into an eddy viscosity-transport equation in a manner identical to that employed in Reference [6]. This model is then combined with the Spalart-Allmaras model to yield an approach designed to predict the characteristics of transitional, three-dimensional, high-speed flows under high disturbance environment conditions. 5

19 . Eddy Viscosity-Transport Equation for Non-Turbulent Fluctuations The framework for the eddy viscosity-transport equation for non-turbulent fluctuation growth is taken from the successful HIDE model [4]. The keys to transforming the model into a usable form for combination with the Spalart-Allmaras one-equation transition/turbulence model are as follows: First, the eddy viscosity definition and non-turbulent energy fluctuation energy equation are presented in Equations (.) and (.). v nt = C k (.) µ τ nt Dk k v k = vntω + +.8vnt Dt τ k x j 3 x j The next step is to multiply both sides of Equation (.) by C µ τ nt. (.) Dk k v k C τnt = C τnt vnt.8vnt C τ µ nt Dt µ Ω + + τ k x j 3 µ (.3) x j The relationship in Equation (.) can now be used to convert Equation (.3) into a transport equation for an eddy viscosity characteristic of non-turbulent fluctuations. Dv v v v = Cµτ v Ω + +.8v Dt x x nt nt nt nt nt nt τ k j 3 j (.4) It should be noted that the form for the diffusion term is not exact. Rather, it is assumed that spatial variations of τ nt are small enough so that the C µ τ nt term may be brought into the spatial derivatives. To complete the equation for the non-turbulent region, the time scale for the HIDE must be defined. This is taken directly from Reference [4] and is shown in Equation (.5). 6

20 τ khide, vnt c5k = c3 + c4γ S+ v v (.5) c = , c = 0.5, c = The Γ term in Equation (.5) is Dhawan and Narasimha s intermittency expression [], which is discussed more thoroughly in the description of the transitional region of the model in Section.4. For simplicity the norm of the strain rate tensor, S, in Equation (.5) is approximated in Equation (.6), where Ω is the vorticity magnitude. Ω S (.6) Equation (.6) can then be inserted into (.5) with the result shown in Equation (.7). τ khide, vnt Ω c5k = c3 + c4γ + (.7) v v The kinetic energy term in Equation (.7) is removed by solving for k in Equation (.) and substituting this expression into Equation (.7): τ khide, vnt Ω c5vnt = c3 + c4γ + v v C µ τ nt (.8) This result can now be placed into Equation (.4) to complete the equation for the nonturbulent part of the one-equation transition/turbulence model: Dv Dt nt = C τ v + x µ nt j nt Ω v v +.8v 3 nt nt c 3 v x nt j v v nt Ω c5vnt + c4γ + vc τ µ nt (.9) The final piece of the non-turbulent section of the model is the formulation of the laminar time scale, τ nt, HIDE. The form for this term is taken from Reference [4] and is shown in Equation (.0) 7

21 τ = c c nt, HIDE Re c * δ c = (.0) = Tu The values defining the Reynolds number in (.0) come from 99% of the boundary layer 8 edge values under any resulting shock produced by the body in the flow. There is more than one method to extract these edge values from the flowfield, and two of these are * discussed thoroughly in Section 3.. The displacement thickness, δ, in Equation (.0) is defined in (.). δ δ * = 0 ρu r dn eu (.) ρ e In Equation (.), n v is the normal distance from the surface and δ is the edge of the boundary layer under any resulting shock..3 Spalart-Allmaras Turbulence Model The Spalart-Allmaras turbulence model used in the creation of the one-equation transition/turbulence model is based on the paper by Spalart and Allmaras [7] and is shown for completeness in the Appendix. Dv~ = C Dt b + σ x ( f ) j t ~~ vω C ( v + v~ ) w f w C κ b t v~ C ~ b v + x j σ x j f v~ d (.) Reading from left to right, the Spalart-Allmaras model (.) has three components: production, destruction, and diffusion (composed of the last two terms). For the high Mach number flows explored in this research, the ratio of turbulent to non-turbulent 8

22 kinematic viscosities is expected to be large enough once the flowfield is entirely turbulent so that the f t function in Equation (.3) can be considered zero. ~ v C t 4 v t t3 = f = C e 0 (.3) This reduces the production and destruction terms of the Spalart-Allmaras turbulence model to Equation (.4), which is used for the turbulent region of the one-equation transition/turbulence model. Dv~ Dt = C b v~~ Ω C w f w v~ d + σ x j ( v + v~ ) v~ C ~ b v + x j σ x j (.4).4 Transition Region Modeling The transition region of the flow is modeled by using a weighted average of the non-turbulent and turbulent eddy viscosity equations through the use of Dhawan and Narasimha s intermittency expression [], modified as shown in Equation (.5). The transition location, s ( ) T Γ ζ = ( ) 0.4ζ ( s, θ ) = e max ( s( θ) st ( θ),0) Re λ λ = 9.0Re 0.75 st ( θ ) (.5) θ, is the distance from the stagnation point on the body to the surface circumferential location of the start of the transition region as dictated by the transition criteria stated in Section.5. Also, s is the surface distance from the stagnation point to any point on the surface of the body. The Reynolds number calculations are defined by 99% of the boundary layer edge values. At surface distances less than the transition location, the intermittency function returns a value of zero, while at surface 9

23 distances far enough from the transition onset location, the function returns a value asymptotically approaching one. In between the transition location and this trailing point, the function returns a monotonically increasing value between zero and one, connecting both of the extremes. Through this process, the transitional region of the final oneequation transition/turbulence model is defined by using the intermittency function to provide a weighted mix of the non-turbulent and turbulent components of the model based on the location along the surface of the test body (See Section.6). Examining Sections. and.3, it is apparent that the diffusion terms in both the non-turbulent and turbulent sections of the model resemble one another. The intermittency function can therefore be used to combine the two components into one single term (.6). v% ΓC b v% + % ν + x j σl σt xj σ x j where Γ Γ = + σ σ 3 l Γ = +.8 Γ σ σ t ( ) (.6) (.7) Looking at (.6) and (.7), it can be seen that when Γ returns a value of 0 (the nonturbulent region of the flow), the diffusion terms in the equation return to the nonturbulent form found in Equation (.4), and when Γ returns a value of (the turbulent region of the flow), the diffusion terms in the equation return to the turbulent form of Equation (.). When Γ returns values between zero and one, a weighted average of the diffusion terms is achieved. 0

24 It was discovered in earlier research [6] that an extra term is required in the transitional region to more accurately model the rise in wall shear stress, or wall heat transfer, that occurs just downstream of transition onset. The result is the term found in Equation (.8), which is added as an additional production term for the eddy viscosity. The constant C t Ct ( ) Γ Γ v% Ω (.8) is set to 0.35, following Reference [6]. Examining the equation, it can be seen that (.8) is active only in the transitional region (0< Γ <)..5 Transition Criteria The criterion for transition of the flow from laminar to turbulent was borrowed from past research work [] and is based on a ratio of turbulent and non-turbulent kinematic viscosities (.9). R T v = % (.9) Cv µ Transition onset is determined by first calculating the maximum value of R T in a boundary layer profile along each axial station over each circumferential ray. Next, a search from the stagnation point aft is done to find the first axial location along each circumferential ray at which the maximum value of R T first exceeds unity. Once this point is found, it is designated the transition onset location, s ( ) T θ, along that ray. If the value of.0 in Equation (.9) is never exceeded along a ray, then the flow is considered laminar over the entire ray, with a value of zero being returned from the intermittency function along the entire ray.

25 .6 lete Transition/Turbulence Model Now that they are established, all of the separate pieces of the one-equation transition/turbulence model can be combined into Equations (.0) and (.) through the use of the intermittency function [] in Equation (.5). Dv~ = Dt ( Γ) + C Γ t + Γ C Cµ τ ntvntω v~ Ω ( Γ) b v~~ Ω C w v v~ d ΓC b v~ + σ x j v~ + + v~ x j σ l σ t x j f w nt c 3 v v nt + c4γ Ω c5vnt + vc τ µ nt (.0) where Γ Γ = + σ σ 3 l Γ = +.8 Γ σ σ t ( ) (.) The model is dependent on the intermittency function to determine whether the equation is simulating non-turbulent, transitional, or turbulent flow. Intermittency values of zero reduce the equation to the eddy viscosity-transport equation for non-turbulent fluctuations found in Section.. Intermittency values of unity reduce the equation to the Spalart- Allmaras turbulence equation found in Section.3. Intermittency values between zero and one produce a continuous weighted average of non-turbulent and turbulent fluctuation modeling in the transitional region of the flow.

26 .7 Calculation of Eddy Viscosity Given the transported quantity v ~ as obtained from Equation (.0), the eddy viscosity as given in Reference [6] is presented in Equation (.). ( ) vt = v % +Γ fv (.) vt = vf % v (.3) When the intermittency function returns a value of unity (i.e. turbulent flow) Equation (.) returns to the purely turbulent Spalart-Allmaras form in Equation (.3). 3 CALCULATION DETAILS 3. Elliptic Cone Grid The experimental test case simulated in this research is Mach 8 flow over an elliptic cone [5]. This case was used in Reference [4] in the study of HIDE-driven transition using the k-ζ transition/turbulence model. The boundaries for the elliptic cone grid are shown in Figure and are based on a sharp-nosed cone with an elliptical cross section of aspect ratio :, and a length of 40 inches (.06 m). The cone half-angle is 7 o along the minor axis. Since the cone is modeled at 0 o angle of attack, the bilateral symmetry of the flowfield requires only one quadrant of the flow to be simulated. Using a cylindrical coordinate system, θ =0 o corresponds to the top centerline, or minor axis endpoint, while θ =90 o corresponds to the leading edge or major axis endpoint., The grid contains 9 grid points in the i-direction (axial coordinate), 65 points in the j-direction (circumferential coordinate), and 73 points in the k-direction (radial coordinate). The distance between the outer radial boundary (k=73) and the surface (k=) was chosen to be large enough so that the grid contains the resulting conical shock, and the flow 3

27 properties along the k=73 boundary are close to freestream conditions. The grid nodes are tightly packed against the surface of the cone, and the spacing between nodes becomes greater as the distance from the surface increases. This is done to capture the boundary layer characteristics around the model more precisely. 3. Determining Boundary Layer Edge Value To determine boundary layer edge properties for use in the equations found in Section, an accurate way of determining the boundary layer edge is first required. For flowfields where shockwaves are not present, the edge of the boundary layer is usually considered to be located at a distance where the local velocity reaches approximately 99% of its freestream value. When shockwaves are present, such techniques are no longer valid due to the discontinuous change in fluid properties across the shock. A conical shock develops in the flowfield discussed in this paper, so a technique had to be devised in order to define the boundary layer edge values. Previously, an inviscid surface data extraction (ISDE) technique had been devised to extract and save surface flow values along the test subject from a converged inviscid run with the code [4]. Using this method, the location of the boundary layer edge does not need to be determined; the surface properties are assumed to be equal to the edge values. Earlier work [4] had found this technique to be successful in finding usable boundary layer edge values and boundary layer displacement values using the edge values in conjunction with a laminar simulation. Initial runs with the present model using this technique resulted in a lower-than-expected non-turbulent time scale. This led to transition onset predictions aft of the experimental location, as shown in Section

28 Therefore, it was determined that a new approach was needed to effectively locate the boundary layer edge, and ultimately, the fluid property values at this location. Profile data from the purely laminar region was extracted from the initial runs of the transition/turbulence code in an effort to search for a better way to locate the boundary layer edge. The u-velocity profile in Figure shows the boundary layer growth, and the conical shock is noticeable. Since the boundary layer edge exists under the conical shock, the pressure profile was then examined in Figure 3 to explore the possibility of locating the edge of the shock as a limit for the search for the boundary layer edge. Even though the grid is not very refined around the conical shock, the discontinuity in the pressure profile is still evident. A search was added to the code to locate the largest pressure gradient along every radial grid line emanating from the cone surface. This point corresponds to the conical shock location. Test cases were run using nodes a set distance (from zero to 4 grid nodes depending on the run) below the shock as the boundary layer edge, but a new problem was detected dealing with the calculation of the displacement thickness, δ *. As the code was progressing towards convergence, negative time scales would start to develop, causing the code to become unstable. Looking at the definition of the time scale in Equation (3.) and knowing that the flow properties such as velocity were being returned as positive values, the displacement thickness was labeled as the problem. τ = (3.) nt, HIDE c Rec * δ Equation (3.) shows the calculation of the displacement thickness and its dependence on the edge values of velocity and density. 5

29 δ δ * = 0 ρu r dn eu (3.) ρ e In a situation where the multiplication of the established edge values of velocity and density are lower than the local product of velocity and density, this part of the integral calculation would yield a negative value. If there is enough of a difference between the two values, the overall integral would return a non-positive value. Checks were performed and it was determined that the axial-momentum ( ρ u ) profile grew in magnitude with increasing distance from the surface in a manner similar to the velocity profile in Figure, but tended to decrease as the flow approached the conical shock as is shown in Figure 4. It was noticed that the maximum value of the axial-momentum profile occurred close to, but slightly under, the conical shock location. This information was used to define the edge of the boundary layer, and ultimately the boundary layer edge values. The method used in the code to locate the boundary layer edge was devised from all of the information described in the preceding paragraphs. First, the conical shock is defined by locating the maximum pressure gradient along each radial grid line emanating from the surface of the cone. This places a roof on the search for the maximum momentum value that is also performed along each radial grid line emanating from the surface of the cone. The location of this maximum for each radial grid is declared the boundary layer edge, and flow properties are extracted from this location for use in the equations in Section. As shown in Section 4.6, this boundary layer edge search (BLES) technique eliminated negative time scales and produced more acceptable results than the ISDE method. 6

30 3.3 Code Description and Boundary Conditions REACTMB [8], an implicit, parallel code for solving general reactive flow problems, was used to perform all runs for this research. The grid was decomposed into 64 equivalent blocks shown in Figure and mapped to 64 processors of the NCSC IBM SP in an effort to reduce the amount of real-world clock time required to achieve convergence. Due to the parallelism of the code, it is necessary to send local boundary layer data to a master processor, which then performs a global search to determine the transition onset locations (See Section.5). The resulting intermittency function is then passed to the other processors. Freestream values are enforced along the incoming flow boundary, surface conditions (no-slip and adiabatic wall) are enforced along the surface, fluid properties are extrapolated at the i max and k max planes, and symmetry conditions are enforced along the j min and j max planes. 3.4 Initialization and Run Parameters Before each run, the entire domain (with the exception of the solid walls) is set to freestream conditions to initiate each run based on the inputted Reynolds number per meter, freestream temperature (K), and freestream Mach number. As per Reference [6], the freestream value of the transported quantity v% is chosen to be v. The code is typically run for 4,000 iterations to achieve a steady-state solution. The CFL number is ramped from 0.5 to 0.0 over the course of the run. Flows at four different Reynolds numbers (See Table ) were simulated to cover the range of transitional data that was compiled experimentally [5] and to help determine the accuracy of the model for a variety of flow conditions. In previous research [4], the average turbulent intensity, Tu, in 7

31 the tunnel during the tests with the elliptic cone was chosen to be.5. Therefore, c =.988 in the non-turbulent time scale of Equation (3.3): τ = c c nt, HIDE Re c * δ c = (3.3) = Tu Reference [4] chooses this value as.0, so in an effort to determine the model s 8 sensitivity to Tu, a set of runs was performed with Tu=.575, which sets c =. Fourteen runs were performed. A purely laminar case and a purely turbulent case with a Reynolds number of.98x0 6 /ft were simulated to compare with transitional runs to ensure that values before and after the transitional region matched the corresponding purely laminar and turbulent solutions. All four studied Reynolds numbers were then run using the ISDE method to define boundary layer edge values along with Tu=.5. Next, all four Reynolds numbers were simulated using the BLES method described in Section 3. along with Tu=.5 to compare the different methods of defining the boundary layer edge values. Lastly, all four Reynolds numbers were simulated using the BLES method and Tu=.575 using the value in Reference [4] for the c constant in order to determine the sensitivity of the model to the choice of Tu. 4 RESULTS AND DISCUSSION 4. erimental Data The experimental data for these runs is obtained from Reference [5]. Transitional data, specifically heat flux at the wall, q w,, was recorded over a wide variety of Reynolds numbers, using the cone described in Section 3.. Pressure and heat transfer gauges were placed in separate quadrants. The heat transfer gauges consist of 4.8mm diameter 8

32 Schmidt-Boelter gauges. These gauges have an uncertainty of ±0% in two-dimensional flows with greater uncertainty in three-dimensional flows. Data was recorded along the 0 o, 45 o, and 88 o circumference locations on the quadrant of the elliptic cone. 4. Calculation of Heat Flux In the results that follow, computed heat fluxes along the cone surface are compared with experimental data. The heat flux at the surface of the cone is calculated using Equation (4.) along the rays that correspond to the circumferential locations where experimental data was recorded. q w µ avgc p Tw = P n r (4.) This computational data can then be plotted along with the experimental heat flux to examine how well the model predicts the transition onset location, extent, and magnitude with respect to the heat flux. 4.3 Convergence As was discussed in Section 3.4, all runs were performed for 4,000 iterations as this converged the turbulence norm to a leveling point. The norm convergence history is shown for the fully laminar and fully turbulent simulations in Figure 5. The turbulent run converges in approximately,00 iterations while the laminar run takes the full 4,000 iterations to reach the same level of convergence. Both runs reach their converged state in a rather linear fashion. The turbulent residual norm is shown in Figure 6 for the Re/L=.98x0 6 /ft transitional runs. The ISDE method converges four orders of magnitude beyond the point where the BLES methods level off. Unlike the linear trends found in the laminar and turbulent convergence curves, the convergence curves for this transitional Reynolds number are jagged and oscillatory. As the transition/turbulence r 9

33 model equations proceed towards an equilibrium, or converged state, grid points near the transition onset locations may shift from non-turbulent to transitional, causing the equation system to relax towards this new equilibrium state. This point movement eventually ceases once the solution reaches its steady state. Figure 7 plots the turbulence norm history for transitional runs with Re/L=.03x0 6 /ft. This time, the BLES method converges two orders of magnitude faster than the ISDE technique. As with the Re/L=.98x0 6 /ft case, spikes in the convergence history occur. The transitional simulations with Re/L=7.8x0 5 /ft are shown in Figure 8. The BLES technique converges five orders of magnitude further than the other technique, and peaks in the curves once again occur. Lastly, Figure 9 shows the turbulence residual norm history with Re/L=6.09x0 5 /ft. The convergence comparison cannot be assessed accurately because the ISDE technique reaches the 4,000 iteration stopping criteria before completely converging. The jagged portions of the convergence curves evident in the other three Reynolds numbers are not as prominent in this set of simulations. This is related to the small region of transitional flow located on the cone (i.e. the majority of the cone is in a purely non-turbulent state). 4.4 Fully Laminar and Turbulent Flow Before any transitional calculations were carried out, fully laminar and a fully turbulent runs at Re/L=.98x0 6 /ft were performed to produce fully laminar and turbulent heat flux curves with which to compare transitional results from the calculations. Figure 0 shows pressure contours at X=0.933 for the laminar calculation. The discontinuity in pressure caused by the conical shock wave is clearly evident. The pressure at the surface is highest at the 90 o position and decreases as θ approaches 0 o. A temperature contour 0

34 plot at the same cross section is shown in Figure revealing a thicker thermal boundary layer for the turbulent simulation. uted heat flux distributions at three circumferential stations are shown in Figure 3 for laminar flow and in Figure 4 for turbulent. As expected, the heat flux curves are higher for the turbulent flow than laminar flow. This explains why in transitional flow, the transition onset location is commonly defined by the minimum position along the heat flux curve, since the heat flux curve will rise to the fully turbulent values within the transitional region. 4.5 Transitional Region The maximum value of R T, the transition criteria found in Equation (.9), at every i-j location is plotted in Figure 5 for the converged transitional run at Re/L=.98x0 6 /ft. When examined with the intermittency function contour plot over the same domain in Figure 6, the dependence of the two equations is apparent. In Figure 5, the darkest region of blue is the location along the cone where the transition criterion has not been met (i.e. R T is less than unity). This results in the intermittency function producing a value of zero, therefore simulating laminar flow. The location where the contour starts to change color signifies the location at which the transition criterion has been met. Correlating this with Figure 6, it is noted that the intermittency function starts to change value at the location where the transition criterion is first met when moving from the leading edge to the trailing edge (left to right in Figure 6). The darkest red in Figure 6 is the region where the intermittency function returns a value of unity, simulating fully turbulent flow. Between the blue and the red regions where the intermittency returns values between zero and unity, the transitional region of the flow is simulated with a mixture of non-turbulent and turbulent flow as discussed throughout

35 Section. Figure 7 confirms this formulation in a body-length cross sectional contour cut of the eddy viscosity. Beyond the distance specified by the transition criteria as the transition onset location, the eddy viscosity grows rapidly close to the surface of the cone as viscous effects start to dominate this region. This growth in eddy viscosity is responsible for the extra fluid heating around the surface in the turbulent flow as described in Section arison of Methods As is discussed in Section 3.3, three different versions of the time scale characteristic of non-turbulent fluctuations were run for each transitional Reynolds number:.98x0 6 /ft,.03x0 6 /ft, 7.8x0 5 /ft, and 6.09x0 5 /ft. The first was done using the ISDE method with an average turbulent intensity of.5. Secondly, a simulation set was performed using the BLES method developed in this paper with an average turbulent intensity of.5. Lastly, the four cases were simulated with the BLES technique and Tu=.575 as is assumed in Reference [4]. These three different versions of the time scale all produce slightly different calculations for the non-turbulent time scale that drives the transition/turbulence model. As was determined in Reference [6], the turbulence model has no real effect in determining the transition onset location. The model for nonturbulent fluctuation growth and its time scale definition are the driving factor for this prediction. Figure 8 plots the non-turbulent fluctuation time scale at X=0.060m versus circumferential angle for all three methods of calculation. As expected, there is a difference between all three versions, with the ISDE method resulting in values below that provided by the two BLES techniques. The highest time scale curve is the boundary layer search method with Tu=.575. The importance of the average turbulent intensity is

36 revealed in this plot. The difference in the time scale constant c in Equation (.0) when using a value of Tu=.5 versus Tu=.575 is only 0.0. Yet, even with such a small difference in the constant, an obvious difference appears in the time scale plot. Figures 9, 0,, and present heat flux distributions for Re/L=.98x0 6 /ft using the three different versions of the time scale. Figures 3, 4, and 5 are similar plots for a Reynolds number of.03x0 6 /ft. Figures 6, 7, and 8 are associated with Re/L=7.8x0 5 /ft and Figures 9, 30 and 3 are with Re/L=6.09x0 5 /ft. Considering the lowest point on the heat flux curves as the point of transition onset, the onset position in Figure 9 for the 45 o curve lags and 88 o curve leads the experimental transition location. Nothing can be inferred about the 0 o curve because the experimental data points are too few to capture the transition process, which occurs very close to the nose of the cone. The extent and magnitude of the transition region along the 45 o curve matches well with the experimental data. The peak heat flux at the 88 o curve is noticeably higher than in the experimental data but nothing can be inferred about the extent of the transitional region at this location since the transitional region extends past the end of the cone. In Figure 0, the transition onset location for the 45 o station closely matches the experimental transition onset location, magnitude and extent while the 88 o curve resembles the same curve in Figure 9, but with an even larger peak value. Oscillations in this curve are associated with pressure oscillations emanating from the conical shock. Figure is similar to the curves found in Figure 0 with the exception that the 88 o and 45 o curves have moved forward slightly. The 45 o curve is still a good approximation of the experimental transition curve. 3

37 To check the validity of the model in the laminar, transitional, and turbulent regions of the flow, the 45 o curve from the last method at Re/L=.98x0 6 /ft is compared against fully laminar and turbulent curves at the same location along the cone in Figure. The model performs well at predicting the heat flux in the laminar and turbulent regions of the flow. The transitional region successfully raises the laminar heat flux up to the turbulent values after peaking around the center of the transitional zone. In Figure 3, corresponding to Re/L=.03x0 6 /ft, the 0 o curve transitions reasonably close to the experimental data, although the fully turbulent level of the heat flux is lower than the experimental value. The 45 o curve lags well behind the experimental data, so much so that the extent and magnitude of transition cannot be compared with experimental data. In accord with experimental data, heat flux predictions at 88 o indicate fully laminar flow for this Reynolds number and for lower values. The curves in Figure 4 better follow the experimental data, with the 0 o curve matching the experimental transition onset location, but the 45 o curve still lags the experimental onset position. Like the ISDE method at 0 o, the BLES technique does not produce a high enough heat flux in the transitional region. Increasing Tu to a value of.575 in Figure 5 moves the transition onset position at 45 o further forward, bringing the prediction in closer agreement with the experimental data. The onset position at 0 o is undisturbed, and laminar flow is still indicated at θ =88 o. Only the 45 o curve transition region magnitude is similar to the data. As with the previous calculations, the magnitude of the heat flux is under-predicted in the transitional region. Similar trends hold for Re/L=7.8x0 5 /ft as seen in Figures 6, 7 and 8. The ISDE technique underestimates the transition onset locations, so much so that the 4

38 calculation predicts that the 45 o curve becomes fully laminar while the experimental data would suggest otherwise. The 0 o curve again under-predicts the magnitude of the heat flux in the transitional region. Figure 7 shows that the use of the BLES method shifts the transition onset locations in Figure 6 forward, but not to the levels indicated in the experimental data. Increasing the average turbulent intensity in Figure 8 brings the onset positions within range of the experimental data, but the calculations still underpredict the heat flux values in the transitional regime. In Figures 9, 30 and 3, corresponding to a Reynolds number of 6.09x0 5 /ft it is clear that none of the variants of the model adequately predict transition onset, placing the onset position further downstream then indicated in the experimental data. The trends evident earlier continue to hold with the ISDE technique predicting transition onset further downstream than the other models. 5 CONCLUSIONS 5. Conclusions The findings of the last section are summarized in Table with respect to each of the time scale calculation methods. The model performs reasonably well for higher Reynolds numbers, but its accuracy starts to deteriorate around the lower end of the spectrum of Reynolds numbers evaluated. Transition onset location was accurately simulated using the third version of the time scale, but the magnitude of the heat flux curves in the transition region of the flowfield is not as accurate. For all transitional runs performed, the 0 o heat flux curve is under-predicted in the transitional region of the code when compared with experimental data, and with respect 5