Session E03: Compressible Flows

Measurements of microscale shock velocity in water droplets using optical imaging and ray tracing analysis

The microscale shock waves induced by laser ablation in liquids are relevant in many applications, including laser surgery and nanoparticle synthesis. Accurate measurements of shock properties, such as shock and particle velocities, are difficult to conduct due to the complexity of optical ablation and the small size of the system. To characterize such shocks, ~60 µm diameter water microdroplets were exposed to focused X-ray laser pulses, which ablate water more controllably and reproducibly than optical lasers. The ablation generated cylindrical shock waves, that were imaged optically inside the drop over a range of time delays. Because the image of the interior of the drop is magnified, we conducted a ray tracing analysis of the optical imaging system. The optical magnification depends on the distance from the center of the drop, being a function rather than a constant factor, and this function depends on the details of the imaging system and on the droplet size and shape. Using the magnification function, we determined the shock positions and velocities inside the drop as a function of time. We also determined the relation between shock and particle velocities by combining these shock velocity measurements with particle velocity measurements in smaller drops, for shocks generated by X-ray pulses with the same energy.   

A modular method for skin friction estimation in compressible boundary layers

The fluid property variation caused by viscous heating implies that empirical skin friction formulas of the form c_f = fcn(Re_θ) are not directly applicable to supersonic boundary layers. We present a method based on the work of Huang et al. (AIAA Journal 1993 31:9, pp. 1600–1604) that estimates the friction and heat transfer coefficients from the Mach number, Reynolds number, and wall temperature. The method is modular in the sense that it works with multiple near-wall scaling formulas ("velocity transformations''), velocity-temperature relationships, and can be calibrated against any incompressible skin friction formula.   

On the Integrability of Compressible Potential Flow in Multiple Spatial Dimensions

This work analyzes the local integrability of the potential equation for unsteady, compressible flow in multiple spatial dimensions. Techniques from coordinate-free differential geometry are applied to cast the compressible potential flow equation as an exterior differential system with an independence condition. The resulting exterior differential system is a linear Pfaffian system which converts the difficult analysis question of local solvability about non-singular points to a simpler problem in linear algebra. The Cartan test is applied to the tableau and torsion tensors from the linear Pfaffian system associated with the potential flow equation to analyze local integrability. The Cartan test gives the conditions needed for which the Cartan-Kähler generalization of the Cauchy-Kowalevski existence theorem may be applied to guarantee that unsteady, compressible, irrotational flow fields have local Taylor series solutions.   

The Big Three: Geometric Unification of the Noh, Sedov, and Guderley Problems

The idealized stagnation shock (formulated by W. Noh in 1983), point blast wave (formulated independently by G. Taylor, L. Sedov, and J. von Neumann in the early 1940s), and symmetric implosion (formulated by K. Guderley in 1942) problems are perhaps the three most famous self-similar shock wave solutions of the gas dynamics equations. These venerable solutions remain indispensable within an incredibly diverse set of physics modeling efforts including supernovae, inertial confinement fusion, or even the ultrasonic destruction of kidney stones. The applicability of these solutions across so large a range of physical scales is itself a natural consequence of the invariance of the gas dynamics equations under scaling transformations, as established classically using the Buckingham-Pi Theorem. In turn, by the 1950s, scale-invariance concepts were successfully integrated within S. Lie's broader group-theory (i.e., geometric, or symmetry) interpretation of differential equations. Accordingly, self-similar solutions of the gas dynamics equations have long since been understood to encode specific realizations of a broader set of underlying symmetries – namely, certain invariant scaling transformations. To this point, this work demonstrates how Lie's symmetry analysis formalism may be employed to rigorously identify both the common basis and distinguishing invariant scaling transformations that give rise to the classical Noh, Sedov-Taylor-von Neumann, and Guderley solutions. In addition to providing a unification framework for these solutions, the symmetry interpretation may also be leveraged to provide a differential geometric view of self-similar shock wave propagation.   

Radially convergent shock waves using a Mie-Grüneisen equation of state

In 1942, Guderley first investigated the problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave that converges in an ideal, inviscid gas. Since then, Guderley type problems have been studied in various disciplines of physics and engineering (i.e., laser fusion and verification activities). Van Dyke and Guttmann provided the seminal solution to a theorized set of initial conditions for the Guderley problem, the 1D radial piston problem. Potential applications of research concerning the 1D radial piston problem have since been limited by Van Dyke and Guttmann’s choice of the ideal gas EOS.

The purpose of this work is to expand the current knowledge base to include a more complex and broadly relevant constitutive law - the Mie-Grüneisen (MG) EOS. The MG EOS models solid materials which resist compression more than their ideal gaseous counterparts. Further, the MG EOS can be reduced to reproduce the results from the Van Dyke and Guttmann investigation. This work uses the solution to the 1D planar piston problem as the first term in a quasi-self-similar series expansion to estimate the solution of the 1D radial piston problem for a material with a non-ideal EOS. Through this analysis, the state-variables in the shocked region of a material are calculated.   

Unsteadiness in a leading-edge separated flow at supersonic speed

Supersonic flow separates across a sharp leading edge with a protrusion present at a downstream length of a few times its height. The resulting separation produces shock-related unsteadiness. The source of the instability is a topic of scientific interest, as any method to reduce or control the unsteady intensity will result in appropriate payload shielding from severe fluid-structure interaction. Spiked body in supersonic flow is a typical case where such a leading-edge flow separation occurs. In the present investigation, a two-dimensional spike fitted in front of a rounded rectangular forebody block is investigated in a supersonic freestream Mach number of M_∞=2.0. Such a configuration is preferred over the axisymmetric counterpart as the former offers flexibility in probing the flow around the spike. Experimentally, oil-flow visualization, high-speed shadowgraph imaging, steady and unsteady wall static pressure measurements are taken using a blow-down supersonic wind tunnel. Complementary two-dimensional computation is performed using the detached-eddy simulation method in a commercial flow solver package. Leading-edge flow separation at supersonic speed produces shocklets along with the separated shear layer. In the considered flow problem, the outer part of the shocklets is propagating along the supersonic flow stream. The other part of the shocklets is trapped inside the recirculation region (observed between the separation region on the spike and the reattachment region on the rounded forebody). Shocklets in the recirculation region are trapped in the subsonic flow region, and hence, travel forth and back between the separation and reattachment zones. The to-and-fro motion of the shocklets perturbs the separation zone and introduces disturbances in the shear layer, which further produces shocklets. The mechanism of self-sustained unsteadiness in a leading-edge separated flow is explained through carefully planned experiments and supplementary computations.   

Mixing enhancement of strut based flame stabilizer for scramjet combustor using passive techniques

In a Supersonic combustion ramjet (Scramjet) engine, the combustion occurs at supersonic velocity as the incoming air remains supersonic. In the scramjet combustor, the mixing and combustion should take place within a few milliseconds. The effect of passive techniques to enhance the mixing characteristics is examined. Two wedges are used downstream of the strut injection to either enhance the mixing due to shock or introduce strong vortices. The first configuration can be named dummy struts (DS) and the second configuration as inverted dummy struts (IDS).  The former configuration enhances mixing due to shock and expansion fans, while the latter enhances mixing due to strong recirculation. Improved delayed detached eddy simulation (IDDES) is a hybrid RANS/LES method used in the current simulation. The validated result of the single strut is compared with the DS and IDS configurations. It is found that the placement of DS plays a vital role. The mixing enhancement was more in IDS, followed by DS. The acoustics characteristics show IDS has the highest noise generation. Dynamic Mode Decomposition (DMD) sheds light on the dominant frequency and corresponding characteristics of waves.   

Hypersonic reacting flow simulations: Toward modular adjoint-based sensitivity analysis

This work presents current progress in hypersonic reacting flow simulations. Hypersonic flows are abundant in aerospace applications and their accurate simulation requires the inclusion of detailed thermochemical models. Choices in the models and numerical methods used are critical for the accuracy, reliability, and the computational cost of the simulations. In this work, a numerical solver for the Navier-Stokes equations based on high-order finite-differences is extended to include finite-rate chemistry effects. This solver is coupled with the open-source library Mutation++ to handle the thermodynamics and chemical kinetics of real gas mixtures. Different numerical approaches for solving the equations are discussed and the influence of the choices of thermophysical models is considered. The results are focused on benchmark cases of flat-plate boundary layers and jet injections in cross-flow. Finally, capabilities for linear-adjoint-based sensitivity analysis are introduced and extended to include finite-rate chemistry and real-gas effects.   

Direct numerical simulation of sonic jet in reacting hypersonic crossflow

At very large Mach numbers, fluid flows are strongly influenced by non-equilibrium gas effects such as finite-rate chemical reactions or internal mode excitation arising from extreme temperatures region. These effects have an order-one influence on quantities of interest, such as stability properties, transition and heating and must be taken into account to achieve effective designs, reliable predictions and successful flow control. 

However, due to challenging flow conditions, most experimental studies are carried out at low temperature to achieve hypersonic speeds and thus remain within the perfect gas regime. Similarly, many simulations have been performed by resorting to simplifying assumptions. There remain, therefore, many open questions regarding the effect of such high-temperatures on the overall behaviour of the flow. In this project, we examine such effects by performing direct numerical simulations of sonic jet injection in a Mach 5 hypersonic crossflow with finite-rate chemistry. The results are then compared to the state-of-the-art modeling approaches (considering perfect gas assumptions) to quantify high temperature effects.   

Not Participating

Experimental studies of supersonic retropropulsion (SRP) dynamics have almost exclusively employed matching gasses for the freestream and retropropulsive jets, hampering our ability to extrapolate these results to conditions more similar to those of engines operating with Earth or Mars’ atmosphere. In this study we explore the influences of gas molecular weight, temperature and ratio of specific heats on SRP standoff distances. Combinations of Helium, Argon, Carbon Dioxide and Nitrogen gasses are explored for single nozzle retrojets with minimal forebody at zero angle of attack at Mach 2. Control volume analysis suggests a dependence of shock standoff distance on the parameter Ψ = MW_∞T_0,j/(MW_j*T_0,∞), and the ratio of specific heats for each gas. The sensitivity of SRP interactions to these parameters is explored for ranges of jet thrust and pressure. The flow is revealed to be a strong function of Ψ, but not the ratio of specific heats. Our results suggest SRP shock standoff distances can be successfully scaled to account for gas composition and flow properties.