Pressure reconstruction of a planar turbulent flow field within a multiply-connected domain with arbitrary boundary shapes

Over the past two decades, it has been demonstrated that the instantaneous spatial pressure distribution in a turbulent flow field can be reconstructed from the pressure gradient field non-intrusively measured by Particle Image Velocimetry (PIV). Representative pressure reconstruction methods include the omnidirectional integration (Liu and Katz, 2006; Liu et al., 2016; Liu and Moreto, 2020), the Poisson equation approach (Violato et al., 2011; De Kat and Van Oudheusden, 2012), the least-square method (Jeon et al., 2015), and most recently, the adjoint-based sequential data assimilation method, which also essentially utilizes the Poisson equation to reconstruct the pressure(He et al., 2020). Most of these previous pressure reconstruction examples, however, were applied to simply-connected domains (Gluzman et al., 2017) only. None of these previous studies have discussed how to apply the pressure reconstruction procedures to a multiply-connected domain (Gluzman et al., 2017). To fill in this gap, this paper presents a detailed report for the first time documenting the implementation procedures and validation results for pressure reconstruction of a planar turbulent flow field within a multiply-connected domain that has arbitrary inner and outer boundary shapes. The pressure reconstruction algorithm used in the current study is the rotating parallelray omni-directional integration algorithm, which, as demonstrated in reference (Liu and Moreto, 2020) based on simply-connected flow domains, offers high-level of accuracy in the reconstructed pressure. While preserving the nature and advantage of the parallel ray omni-directional pressure reconstruction at places with flow data, the new implementation of the algorithm is capable of processing an arbitrary number of inner void areas with arbitrary boundary shapes. Validation of the multiply-connected domain pressure reconstruction code is conducted using the DNS (Direct Numerical Simulation) isotropic turbulence field available at the Johns Hopkins Turbulence Databases, with 1000 statistically independent pressure gradient field realizations embedded with random noise used to gauge the code performance. For further validation, the code is also applied for pressure reconstruction from the DNS pressure gradient in the ambient flow field of a shock-induced non-spherical bubble collapse in water (Johnsen and Colonius, 2009). The successful implementation of the parallel ray pressure reconstruction method to multiply-connected domains paves the way for a variety of important applications including, for example, experimental characterization of pressure field changes during the process of cavitation bubble inception, growth and collapse, non-intrusive unsteady aerodynamic force assessment for an arbitrary body shape immersed in flows, and multi-phase flow investigations, etc. In particular, as an immediate follow-up effort, the parallel ray pressure code will be used for the instantaneous pressure distribution reconstruction of the turbulent flow surrounding cavitation inception bubbles occurring on top of a cavity trailing corner based on high-speed PIV measurements.

On Subharmonic and Entire Functions of Small Order: After Kjellberg

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

ON NONHOMOGENEOUS BOUNDARY VALUE PROBLEM FOR THE STATIONARY NAVIER-STOKES EQUATIONS IN A SYMMETRIC CUSP DOMAIN

The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.

Construction of complex potentials for multiply connected domain

The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential – an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given 𝑛-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and 𝑛 logarithmic summands. The method is easily computable.

Connectivity of dynamic graphs in multiply connected spaces

Dynamic geometric graphs are natural mathematical models of many real-world systems placed and moving in space: computer ad hoc networks, transport systems, territorial distributed systems for various purposes. An important property of such graphs is connectivity, which is difficult to maintain during movement due to the presence of obstacles on the ground. In this paper, a model of a multiply connected region with obstacles of the “city blocks” type is constructed and the behavior of the characteristics of dynamic graphs located in such domains is studied. A probabilistic approach to the study of graphs is proposed, in which their characteristics are considered as random processes. For graphs of different scales, dependences of the connectivity probability, the number of components on the parameters of a multiply connected region, and the radius of stable signal reception / transmission were found. The mathematical expectation of the number of components in the starting random geometric graph is found. The significant influence not only of geometrical parameters, but also of the topological characteristics of a multiply-connected domain has been revealed. Graphs of changes in the probability of connectedness of a dynamic graph over time are constructed on the basis of calculating the average value over the set of realizations of the random process of moving network nodes. They are characterized by a periodic component that correlates with the structure of a multiply connected region, and a component that exponentially decreases with time. The dependence of the probability of connectedness of the graph on the direction of the network displacement vector was studied, which turned out to be very significant. The results obtained give an idea of the influence of a multiply-connected domain on the dynamics of graphs, and can be used in control algorithms for mobile distributed systems to ensure their spatial connectivity.

Dirichlet spaces of domains bounded by quasicircles

Consider a multiply-connected domain [Formula: see text] in the sphere bounded by [Formula: see text] non-intersecting quasicircles. We characterize the Dirichlet space of [Formula: see text] as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary values. Thereafter, we define a Grunsky operator on direct sums of Dirichlet spaces of the disk, and give a second characterization of the Dirichlet space of [Formula: see text] as the graph of the generalized Grunsky operator in direct sums of the space [Formula: see text] on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.