## Wallershtein, Idan

##### PhD student
Email
wallersh@post.bgu.ac.il
Office
54/319
Website
https://physics.bgu.ac.il/~wallersh/
Researcher identification
ORCID

## Education

• 2022- now, PhD Title is not available
with Eytan Grosfeld
• 2016- 2022, MSc New insights to the potential flows around axisymmetric bodies
with Uri Keshet and Eytan Grosfeld

Abstract/Description: We derive new analytical and semi-analytical results for potential flows around axisymmetric bodies. Starting with the well-known problem of an incompressible potential flow around a 2D disk or a 3D sphere, we study generalizations involving an arbitrary number of dimensions, non-spherical bodies, compressible effects, and the transition to the supersonic regime.

Inviscid, incompressible flows are analytically known only around a handful of simple bodies, in particular hyperspheres. We expand the list of flows for which the potential and the body shape can be simultaneously expressed as finite combinations of elementary functions. These body shapes are solutions to polynomial equations of the radial coordinate $r$, with coefficients depending on the polar angle $\theta$. We analyze the associated flow properties, including the emergence of stagnation cusps on some bodies.

Compressibility effects are examined for polytropic fluids (of adiabatic index $\gamma$) around d-dimensional hyperspheres. We generalize the Janzen-Rayleigh expansion (JRE) of the potential in the incident Mach number squared, $\mach_\infty^2$, which was until now limited to $\mach_\infty^4$ order in 3D. We show that each term in the JRE combines products not only of powers of $r$ and Jacobi polynomials $J(\cos\theta)$, but also power of $\ln(r)$, thus extending the JRE to arbitrary order. We demonstrate and conjecture that for any $\gamma$, logarithmic terms are absent in 2D, but appear in 3D with a power incremented every $\mach_\infty^6$ order.

We demonstrate how an arbitrary-order JRE can be used to study compressibility effects, focusing on the potential flow in front of an axisymmetric body, which plays an important role in various physical problems. We thus improve a hodographic approximation for the flow in front of an axisymmetric 3D body. The flow is then generalized for arbitrary prolate spheroids. To this end, we develop a pseudospectral numerical code and discover a simple, approximate scaling, relating the flows in front of different prolate spheroids, independent of $\mach_\infty$ and $\gamma$.

Finally, we present a new attack on the transonic controversy: can shock-free potential flows exceed the critical Mach number $\mach_{cr}$, i.e. reach supersonic velocities anywhere on the body.
With an 80-term JRE for the 2D flow of a diatomic gas ($\gamma=7/5$) around a disk, we impose constraints stronger than previously available.
Using non-linear sequence transformations, we obtain accurate results for $\mach_{cr}$ (13 digit accuracy), and for the cavity Mach number $\mach_{ca}$ (five-digit accuracy) which yields a locally vanishing sound velocity.
Ruling out singularities of an algebraic, confluent, or exponential type, our marginal results indicate that a different analytic approach is necessary to resolve the controversy.

### Publications

1. Wallerstein, I.S. and Keshet, U., "Compressible Potential Flows Around Round Bodies: Janzen–Rayleigh Expansion Inferences", J. Fluid Mech. 932, A6 (2021)