Understanding the process of transition to turbulence has important implications for the design of heat shields for hypersonic re-entry or cruise vehicles. Turbulent flow causes a higher heat flux compared to laminar flow, and insufficient thermal protection may be catastrophic for the vehicle.

Roughness-induced boundary-layer transition

A localized 3-D [1] or 2-D [2] roughness element inside the boundary layer on the surface of such a vehicle may profoundly alter the instability of the boundary layer and could lead to an early onset of turbulent flow. However, our understanding of this alteration is far from comprehensive.

Direct numerical simulations of laminar-turbulent transition in a supersonic boundary layer can improve our understanding. In the figure, vortical structures developing behind a square roughness located at x=15 are visualized (q-criterion colored by the distance from the wall) for a Ma=4.8 flow [1].

[1] O. Marxen, G. Iaccarino, E.S.G. Shaqfeh (2011). AIAA paper 2011-0567, doi
[2] O. Marxen, G. Iaccarino, E.S.G. Shaqfeh (2010). J. Fluid Mech. 648: 435–469, doi

Chemically-reacting gases for equilibrium and non-equilibrium conditions

The temperature in the boundary layer on the surface of a hypersonic vehicle is often very high. For such a high temperature, dissociation reactions occur in the gas mixture, changing the thermodynamic and transport properties. Due to a lack of suitable methods, direct numerical simulations of boundary-layer instability and laminar-turbulent transition are routinely carried out only if the gas properties are assumed to be constant (non-reacting calorically perfect gas).

We have developed a numerical method which couples a solver for the unsteady Navier–Stokes equations with a computational library providing thermodynamic and transport properties for a gas in thermo-chemical equilibrium [3] or non-equilibrium [4]. This allows us to compute boundary-layer profiles for the atmosphere of the Earth (dashed lines) and Mars (solid lines), as shown in the figure for equilibrium conditions at Ma=10, and to investigate their instability.

[3] O. Marxen, T. Magin, G. Iaccarino, E. Shaqfeh (2011). Phys. Fluids 23: 084108, doi
[4] O. Marxen, G. Iaccarino, E.S.G. Shaqfeh (2010). Annual Research Briefs, Center for Turbulence Research, Stanford University

Modern laminar airfoils for the wings of unmanned aerial vehicles or wind-turbine blades possess a pressure distribution in which a long, weakly favorable pressure gradient is followed by a strong adverse pressure gradient. The latter causes the initially laminar boundary layer to separate from the wall. Laminar–turbulent transition occurs in the detached shear layer causing reattachment in the mean, and a closed laminar separation bubble (LSB) is formed. LSBs may strongly affect lift and drag as well as noise generation.

Instability mechanisms and laminar-turbulent transition

Several different instability mechanisms are active in the detached shear layer [1-4]. During transition the shear layer rolls up to form spanwise vortices [4] (visualized by the spanwise vorticity in the figure). These large-scale structures often quickly break up into small-scale vortices [3] before a turbulent reattached boundary layer develops. In the figure, the time-averaged reattachment location is marked by an arrow.

[1] O. Marxen, M. Lang, U. Rist (2012). J. Fluid Mech. 711: 1–26, doi
[2] O. Marxen, M. Lang, U. Rist, O. Levin, D. Henningson (2009). J. Fluid Mech. 634: 165–189, doi
[3] O. Marxen, U. Rist, S. Wagner (2004). AIAA J. 42 (5): 937–944, doi
[4] O. Marxen, M. Lang, U. Rist, S. Wagner (2003). Flow Turbulence Comb. 71: 133–146, doi

Viscous-inviscid interaction, bubble bursting, and flow control

A feedback loop with viscous–inviscid interaction as an essential ingredient occurs during laminar-turbulent transition in a laminar separation bubble: the mean flow in the rear part of the bubble changes as a result of transition, which in turn provokes a global change in the pressure distribution. The boundary layer reacts to the change in pressure in such a way that the separation bubble shirnks and the separation point moves downstream. The distance of the shear layer from the wall diminishes and traveling instability waves are thus less amplified in the laminar part of the bubble [5,6]. This effect can be employed for the control of separation bubbles [7].

The so-called bubble bursting is related to the interaction effect described above. During bubble bursting, a small change in a governing parameter causes a short bubble to become much longer. In our case, this parameter is the amplitude of a boundary-layer perturbation which triggers transition [2]. Vortex shedding is still observable (arrows in the figure).

[1] O. Marxen, D.S. Henningson (2011). J. Fluid Mech. 671: 1–33
[2] O. Marxen, U. Rist (2010). J. Fluid Mech. 660: 37–54
[3] R. Kotapati, R. Mittal, O. Marxen, F. Ham, D. You, L.N. Cattafesta III (2010). J. Fluid Mech. 654: 65–97

Human upper airways

The flow in human airways, specifically in the larynx or in the trachea, may be laminar, transitional, or turbulent during the inhalation cycle. Interest in using the mouth or nose as portals for the delivery of medicine is growing rapidly. In order to correctly predict particle deposition during this therapeutic drug aerosol administration, a better characterization of the flow is required.

We perform 3-D [1] and 2-D [2] numerical simulations of the flow through a channel, which serves as a simple model of an airway segment. Time-dependent inflow profiles are applied to model the change in flow speed during breathing. The natural variability of the breathing process is captured by means of uncertainty quantification using stochastic collocation.

Laminar-turbulent transition is investigated for a case with a geometrical obstruction at the airway wall [1], which could occur due to e.g. a tumor. The occurrence of transition is associated with an increase in RMS values. For the case depicted in the figure, transition starts at around t/T0=12 and ends at t/T0=26. The corresponding variability is represented by error bars. The figure shows that variability for both mean flow and RMS values is largest at times when transition occurs.

[1] O. Marxen, T. Magin (2012). Bulletin of the American Physical Society (65th Annual Meeting of the APS Division of Fluid Dynamics), Vol. 57 (17) (APS abstract)
[2] O. Marxen (2011). Bulletin of the American Physical Society (64th Annual Meeting of the APS Division of Fluid Dynamics, 2011), vol. 56 (APS abstract)

High-order numerical methods for the Navier-Stokes equations

We have developed a new method [1] that allows us to compute steady solutions of the Navier-Stokes equations for configurations that are globally unstable. A dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations is added to the right-hand side of the Navier-Stokes equations, damping the unstable temporal frequencies. The method is easy to implement for existing solvers and converges quickly to a state with a very small residual (figure).

[1] E. Åkervik, L. Brandt, D. S. Henningson, J. Hœpffner, O. Marxen, P. Schlatter (2006). Phys. Fluids 18, 068102, doi

Uncertainty quantification

A non-deterministic analysis based on non-intrusive stochastic collocation is performed for the problem of supersonic boundary-layer instability with a 2-D roughness element [2]. The height of the roughness element h is not known exactly, but only in a statistical sense: we assume a certain probability density function (pdf in the upper left corner of the figure). For several heights within this range, the additional amplification caused by the presence of the roughness, the growth modifier Delta N, has been computed to obtain the response curve Delta N as a function of h (upper right corner). In a final step, Monte Carlo sampling is used to obtain the resulting pdf for Delta N.

[2] O. Marxen, G. Iaccarino, E.S.G. Shaqfeh. “Linear and non-linear disturbance evolution in a compressible boundary-layer with localized roughness”, In 7th IUTAM Symposium on Laminar-Turbulent Transition (ed. P. Schlatter, D. Henningson), IUTAM Bookseries 18, Springer, pp. 271-276, doi

High-performance computing

A code for the direct numerical simulations has been optimized in collaboration with the High Performance Computing Center Stuttgart (HLRS). In particular, F. Svensson worked on improving the code and executed performance tests. The following picture and text are taken from [3].

The sustained performance on 70 nodes (8 CPUs per node) of a NEC SX-8 vector super computer at the High Performance Computing Center Stuttgart (HLRS) reached 2.6 Tflop/s using a large test case with 1100 million grid points [1]. The strong scaling plot (figure) between 15 and 70 nodes, show an efficiency between 39% and 30% (gray). A smaller test case with 314 million grid points reached 1.4 Tflop/s on 30 nodes. The single node performance reached 59 Gflop/s and an efficiency of 46%.

[3] S.R. Tiyyagura, P. Adamidis, R. Rabenseifner, P. Lammers, S. Borowski, F. Lippold, F. Svensson, O. Marxen, S. Haberhauer, A.P. Seitsonen, J. Furthmüller, K. Benkert, M. Galle, T. Bönisch, U. Küster, M.M. Resch (2008). “Teraflops sustained performance with real world applications”, International Journal of High Performance Computing Applications 22 (2), pp. 131–14, doi