Adjoint-based linear sensitivity of a supersonic boundary layer to steady wall blowing-suction/heating-cooling. (English) Zbl 07785768
Summary: For a Mach 4.5 flat-plate adiabatic boundary layer, we study the sensitivity of the first, second Mack modes and streaks to steady wall-normal blowing/suction and wall heat flux. The global instabilities are characterised in frequency space with resolvent gains and their gradients with respect to wall-boundary conditions are derived through a Lagrangian-based method. The implementation is performed in the open-source high-order finite-volume code BROADCAST and algorithmic differentiation is used to access the high-order state derivatives of the discretised governing equations. For the second Mack mode, the resolvent optimal gain decreases when suction is applied upstream of Fedorov’s mode \(S\)/mode \(F\) synchronisation point, leading to stabilisation, and the converse when applied downstream. The largest suction gradient is in the region of branch I of mode \(S\) neutral curve. For heat-flux control, strong heating at the leading edge stabilises both the first and second Mack modes, the former being more sensitive to wall-temperature control. Streaks are less sensitive to any boundary control in comparison with the Mack modes. Eventually, we show that an optimal actuator consisting of a single steady heating strip located close to the leading edge manages to damp the linear growth of all three instability mechanisms.
MSC:
| 76E05 | Parallel shear flows in hydrodynamic stability |
| 76E19 | Compressibility effects in hydrodynamic stability |
| 76N25 | Flow control and optimization for compressible fluids and gas dynamics |
| 76N20 | Boundary-layer theory for compressible fluids and gas dynamics |
| 76J20 | Supersonic flows |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
Keywords:
flat-plate adiabatic boundary layer; boundary layer control; boundary layer stability; global instability analysis; high-order finite volume code BROADCAST; algorithmic differentiationReferences:
| [1] | Airiau, C., Bottaro, A., Walther, S. & Legendre, D.2003A methodology for optimal laminar flow control: application to the damping of Tollmien-Schlichting waves in a boundary layer. Phys. Fluids15 (5), 1131-1145. · Zbl 1186.76015 |
| [2] | Amestoy, P., Duff, I., L’Excellent, J.-Y. & Koster, J.2001A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix. Anal. Applics.23 (1), 15-41. · Zbl 0992.65018 |
| [3] | Balakumar, P. & Hall, P.1999Optimum suction distribution for transition control. Theor. Comput. Fluid Dyn.13 (1), 1-19. · Zbl 0943.76026 |
| [4] | Balay, S., et al.2019 PETSc users manual. ANL-95/11. |
| [5] | Barbagallo, A., Dergham, G., Sipp, D., Schmid, P.J. & Robinet, J.-C.2012Closed-loop control of unsteadiness over a rounded backward-facing step. J. Fluid Mech.703, 326-362. · Zbl 1248.76051 |
| [6] | Batista, A. & Kuehl, J.2020Local wall temperature effects on the second-mode instability. J. Spacecr. Rockets57 (3), 580-595. |
| [7] | Boujo, E.2021Second-order adjoint-based sensitivity for hydrodynamic stability and control. J. Fluid Mech.920, A12. · Zbl 1503.76031 |
| [8] | Brandt, L., Sipp, D., Pralits, J. & Marquet, O.2011Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech.687, 503-528. · Zbl 1241.76340 |
| [9] | Bugeat, B., Chassaing, J.-C., Robinet, J.-C. & Sagaut, P.20193D global optimal forcing and response of the supersonic boundary layer. J. Comput. Phys.398, 108888. · Zbl 1453.76194 |
| [10] | Chanteux, X., Bégou, G., Deniau, H. & Vermeersch, O.2022 Construction and application of transition prediction databased method for 2nd mode on sharp cone. AIAA AVIATION 2022 Forum. |
| [11] | Chu, B.-T.1965On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica1 (3), 215-234. |
| [12] | Cinnella, P. & Content, C.2016High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows. J. Comput. Phys.326, 1-29. · Zbl 1373.76058 |
| [13] | Crivellini, A. & Bassi, F.2011An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations. Comput. Fluids50 (1), 81-93. · Zbl 1271.76164 |
| [14] | Dalcin, L., Paz, R., Kler, P. & Cosimo, A.2011Parallel distributed computing using Python. Adv. Water Resour.34 (9), 1124-1139. |
| [15] | Fedorov, A. & Khokhlov, A.2002Receptivity of hypersonic boundary layer to wall disturbances. Theor. Comput. Fluid Dyn.15 (4), 231-254. · Zbl 1082.76069 |
| [16] | Fedorov, A., Ryzhov, A., Soudakov, V. & Utyuzhnikov, S.2014Numerical simulation of the effect of local volume energy supply on high-speed boundary layer stability. Comput. Fluids100, 130-137. |
| [17] | Fedorov, A. & Tumin, A.2011High-speed boundary-layer instability: old terminology and a new framework. AIAA J.49 (8), 1647-1657. |
| [18] | Fong, K.D., Wang, X. & Zhong, X.2014Numerical simulation of roughness effect on the stability of a hypersonic boundary layer. Comput. Fluids96, 350-367. · Zbl 1391.76279 |
| [19] | Gaster, M.1974On the effects of boundary-layer growth on flow stability. J. Fluid Mech.66 (3), 465-480. · Zbl 0296.76021 |
| [20] | George, K.J. & Sujith, R.I.2011On Chu’s disturbance energy. J. Sound Vib.330 (22), 5280-5291. |
| [21] | Guo, P., Gao, Z., Jiang, C. & Lee, C.-H.2021Sensitivity analysis on supersonic-boundary-layer stability subject to perturbation of flow parameters. Phys. Fluids33 (8), 084111. |
| [22] | Hanifi, A., Schmid, P. & Henningson, D.1996Transient growth in compressible boundary layer flow. Phys. Fluids8 (3), 826-837. · Zbl 1025.76536 |
| [23] | Hascoet, L. & Pascual, V.2013The tapenade automatic differentiation tool: principles, model, and specification. ACM Trans. Math. Softw.39 (3), 1-43. · Zbl 1295.65026 |
| [24] | Hernández, V., Román, J., Tomás, A. & Vidal, V.2007 Krylov-Schur methods in SLEPc. Universitat Politecnica de Valencia. Tech. Rep. STR-7. |
| [25] | Huang, Z. & Wu, X.2016 The effect of local steady suction on the stability and transition of boundary layer on a flat plate. In 8th AIAA Flow Control Conference, p. 3471. |
| [26] | Huerre, P. & Monkewitz, P.A.1990Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech.22 (1), 473-537. · Zbl 0734.76021 |
| [27] | Joslin, R.1998 Overview of laminar flow control. NASA Tech. Rep. TP-1998-20705. |
| [28] | Kazakov, A., Kogan, M. & Kuparev, V.1995Optimization of laminar-turbulent transition delay by means of local heating of the surface. Fluid Dyn.30 (4), 563-570. · Zbl 0875.76491 |
| [29] | Ma, Y. & Zhong, X.2003Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions. J. Fluid Mech.488, 31-78. · Zbl 1063.76544 |
| [30] | Mack, L.1963The inviscid stability of the compressible laminar boundary layer. Space Prog. Summary37, 23. |
| [31] | Mack, L.1993 Effect of cooling on boundary-layer stability at Mach number 3. In Instabilities and Turbulence in Engineering Flows (ed. D.E. Ashpis, T.B. Gatski & R. Hirsh), pp. 175-188. Springer. |
| [32] | Malik, M.1989Prediction and control of transition in supersonic and hypersonic boundary layers. AIAA J.27 (11), 1487-1493. |
| [33] | Marquet, O., Sipp, D. & Jacquin, L.2008Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech.615, 221-252. · Zbl 1165.76012 |
| [34] | Martínez-Cava, A.2019 Direct and adjoint methods for highly detached flows. PhD thesis, Espacio. |
| [35] | Martinez-Cava, A., Chávez-Modena, M., Valero, E., De Vicente, J. & Ferrer, E.2020Sensitivity gradients of surface geometry modifications based on stability analysis of compressible flows. Phys. Rev. Fluids5 (6), 063902. |
| [36] | Masad, J.1995Transition in flow over heat-transfer strips. Phys. Fluids7 (9), 2163-2174. · Zbl 1027.76543 |
| [37] | Masad, J. & Abid, R.1995On transition in supersonic and hypersonic boundary layers. Intl J. Engng Sci.33 (13), 1893-1919. · Zbl 0899.76149 |
| [38] | Mckeon, B.J. & Sharma, A.S.2010A critical-layer framework for turbulent pipe flow. J. Fluid Mech.658, 336-382. · Zbl 1205.76138 |
| [39] | Mettot, C.2013 Linear stability, sensitivity, and passive control of turbulent flows using finite differences. PhD thesis, Palaiseau, Ecole polytechnique. |
| [40] | Mettot, C., Renac, F. & Sipp, D.2014Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: application to open-loop control. J. Comput. Phys.269, 234-258. · Zbl 1349.76129 |
| [41] | Miró Miró, F. & Pinna, F.2018Effect of uneven wall blowing on hypersonic boundary-layer stability and transition. Phys. Fluids30 (8), 084106. · Zbl 1460.76600 |
| [42] | Morkovin, M.1994Transition in open flow systems-a reassessment. Bull. Am. Phys. Soc.39, 1882. |
| [43] | Nibourel, P., Leclercq, C., Demourant, F., Garnier, E. & Sipp, D.2023Reactive control of second mack mode in a supersonic boundary layer with free-stream velocity/density variations. J. Fluid Mech.954, A20. · Zbl 1527.76061 |
| [44] | Oz, F., Goebel, T.E., Jewell, J.S. & Kara, K.2023Local wall cooling effects on hypersonic boundary-layer stability. J. Spacecr. Rockets60 (2), 412-426. |
| [45] | Park, J. & Zaki, T.2019Sensitivity of high-speed boundary-layer stability to base-flow distortion. J. Fluid Mech.859, 476-515. · Zbl 1415.76281 |
| [46] | Poinsot, T. & Lele, S.1992Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys.101 (1), 104-129. · Zbl 0766.76084 |
| [47] | Poulain, A., Content, C., Sipp, D., Rigas, G. & Garnier, E.2023Broadcast: a high-order compressible CFD toolbox for stability and sensitivity using algorithmic differentiation. Comput. Phys. Commun.283, 108557. · Zbl 07693410 |
| [48] | Pralits, J., Airiau, C., Hanifi, A. & Henningson, D.2000Sensitivity analysis using adjoint parabolized stability equations for compressible flows. Flow Turbul. Combust.65 (3), 321-346. · Zbl 1094.76513 |
| [49] | Pralits, J., Hanifi, A. & Henningson, D.2002Adjoint-based optimization of steady suction for disturbance control in incompressible flows. J. Fluid Mech.467, 129-161. · Zbl 1042.76021 |
| [50] | Reed, H. & Nayfeh, A.1986Numerical-perturbation technique for stability of flat-plate boundary layers with suction. AIAA J.24 (2), 208-214. · Zbl 0595.76100 |
| [51] | Reynolds, G. & Saric, W.1986Experiments on the stability of the flat-plate boundary layer with suction. AIAA J.24 (2), 202-207. |
| [52] | Rigas, G., Sipp, D. & Colonius, T.2021Nonlinear input/output analysis: application to boundary layer transition. J. Fluid Mech.911, A15. · Zbl 1461.76223 |
| [53] | Roman, J., Campos, C., Romero, E. & Tomás, A.2015 SLEPc users manual. D. Sistemes Informàtics i Computació Universitat Politècnica de València, Valencia, Spain, Rep. No. DSIC-II/24/02. |
| [54] | Saint-James, J.2020 Prévision de la transition laminaire-turbulent dans le code elsa. extension de la méthode des paraboles aux parois chauffées. PhD thesis, Institut Supérieur de l’Aéronautique et de l’Espace (ISAE). |
| [55] | Schmid, P.J., Henningson, D.S. & Jankowski, D.F.2002Stability and Transition in Shear Flows. Applied Mathematical Sciences, Appl. Mech. Rev.142, B57-B59. |
| [56] | Sciacovelli, L., Passiatore, D., Cinnella, P. & Pascazio, G.2021Assessment of a high-order shock-capturing central-difference scheme for hypersonic turbulent flow simulations. Comput. Fluids230, 105134. · Zbl 1521.76584 |
| [57] | Shen, Y., Zha, G. & Chen, X.2009High order conservative differencing for viscous terms and the application to vortex-induced vibration flows. J. Comput. Phys.228 (22), 8283-8300. · Zbl 1422.76138 |
| [58] | Sidorenko, A., Gromyko, Y., Bountin, D., Polivanov, P. & Maslov, A.2015Effect of the local wall cooling/heating on the hypersonic boundary layer stability and transition. Prog. Flight Phys.7, 549-568. |
| [59] | Sipp, D. & Marquet, O.2013Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn.27 (5), 617-635. |
| [60] | Soudakov, V., Fedorov, A. & Egorov, I.2015Stability of high-speed boundary layer on a sharp cone with localized wall heating or cooling. Prog. Flight Phys.7, 569-584. |
| [61] | Stuckert, G., Lin, N. & Herbert, T.1995 Nonparallel effects in hypersonic boundary layer stability. In 33rd Aerospace Sciences Meeting and Exhibit, p. 776. |
| [62] | Sutherland, W.1893LII. The viscosity of gases and molecular force. Lond. Edinb. Dublin Philos. Mag. J. Sci.36 (223), 507-531. · JFM 25.1544.01 |
| [63] | Towne, A., Rigas, G., Kamal, O., Pickering, E. & Colonius, T.2022Efficient global resolvent analysis via the One-Way Navier-Stokes equations. J. Fluid Mech.948, A9. · Zbl 1516.76026 |
| [64] | Walther, S., Airiau, C. & Bottaro, A.2001Optimal control of Tollmien-Schlichting waves in a developing boundary layer. Phys. Fluids13 (7), 2087-2096. · Zbl 1184.76581 |
| [65] | Wang, X. & Zhong, X.2009Effect of wall perturbations on the receptivity of a hypersonic boundary layer. Phys. Fluids21 (4), 044101. · Zbl 1183.76562 |
| [66] | Wang, X., Zhong, X. & Ma, Y.2011Response of a hypersonic boundary layer to wall blowing-suction. AIAA J.49 (7), 1336-1353. |
| [67] | Wang, Y., Ferrer, E., Martínez-Cava, A., Zheng, Y. & Valero, E.2019Enhanced stability of flows through contraction channels: combining shape optimization and linear stability analysis. Phys. Fluids31 (7), 074109. |
| [68] | Zhao, R., Wen, C., Tian, X., Long, T. & Yuan, W.2018Numerical simulation of local wall heating and cooling effect on the stability of a hypersonic boundary layer. Intl J. Heat Mass Transfer121, 986-998. |
| [69] | Zuccher, S., Luchini, P. & Bottaro, A.2004Algebraic growth in a Blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. J. Fluid Mech.513, 135-160. · Zbl 1107.76032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
