Session W16: General Fluid Dynamics: Mathematical Methods (10:00am - 10:45am CST)

Bounds on flow rate and friction factor in pressure-driven flow through helical pipes

We obtain a lower bound on the flow rate and an equivalent upper bound on the friction factor for flow through a helical pipe driven by a pressure differential in the limit of high Reynolds number using the well-known background method. We obtain these bounds as a function of geometrical parameters, i.e., the curvature and the torsion of the pipe, by considering a two-dimensional background flow with varying boundary layer thickness along the circumference of the pipe. We compare our findings with available experimental data. Finally, we present a sufficient criterion for the applicability of the background flow to pressure-driven flow and surface-velocity-driven flow problems.   

Direct statistical simulation: an alternative approach to turbulence

\\ Direct statistical simulation (DSS) is a revolutionary mathematical framework in the field of computational fluid dynamics. DSS is especially suited for simulating the large scale motions of the inhomogeneous and anisotropic turbulent flows, e.g., Tobias etal, (2011). In the DSS framework, the statistics, namely cumulant, are employed to describe the fluid motions of different scales. The large scale behaviour of the fluid that correspond to the low order cumulant terms are therefore smooth in phase space and need much fewer degrees of freedom in both space and time for the numerical computation as that required for the direct numerical simulation (DNS). DSS is expected to compute the turbulent flow in the extreme turbulent regimes beyond the reach of DNS. \\ We present a matrix-free method for computing the governing cumulant equations up to the third order and demonstrate its numerical supremacy via the Barotropic jets problem. For this illustrative 2-dimensional problem, we observe that our matrix-free method is at least $10^3$ to $10^4$ times faster than the DNS computation or the conventional matrix-based algorithm of DSS with the same numerical accuracy.   

A Reduced-Physics Model of Hele-Shaw Flow Using Asymptotic Basis Functions

Asymptotic basis functions (ABF) provide a model-driven source of basis functions that can be used in Galerkin projection to form a reduced-physics model (RPM) of complex physical phenomena. The ABF are obtained from an asymptotic analysis, in which each order of the asymptotic expansion, including the corresponding gauge function, gives the ABFs. Hele-Shaw flow provides an interesting and challenging model problem for computing the dynamics of interfacial flows with complex interface topologies that span both linear and nonlinear regimes and includes stable and unstable interface dynamics depending on the time evolution of the gap width. The first several terms in an expansion of the evolution equations for Hele-Shaw flow in the limit of small interface perturbation are used in a Galerkin projection to form a RPM that incorporates both the linear and nonlinear behavior in this limit. Galerkin projection allows for application of the RPM to $O(1)$ values of the interface perturbation length scale that occur over longer evolution times. This approach provides a general framework for the development of physics-based RPM in both regular and singular-perturbation problems, thereby extending their applicability to a broader parameter space.   

Mixing and transport in the presence of a source

A common topic in mixing is the initial value problem \begin{equation*} \partial_t \theta + \bm{u}(\bm{x},t) \cdot \nabla \theta = D \Delta \theta + s(\bm{x},t) \end{equation*} where $\bm{u}$ is an incompressible flow stirring the mean-zero passive scalar concentration $\theta$ and $s$ is a mean-zero internal source. The initial concentration is unmixed and, for $s = 0$, will relax to the zero steady state. This convergence is typically maximized by a flow that increases the gradients of the passive scalar, dissipating variance via molecular diffusion. For $s \neq 0$, the passive scalar still relaxes to a steady state, but variance can now also be reduced --- sometimes optimally [3] depending on the distribution of the source --- by transporting hot spots to cold spots. We compare the optimality of gradient production to transport for various source distributions by employing multiscale asymptotics [1,2] and observing that the two mechanisms decouple when the length scale of the source is strongly separated from the length scale of the gradient producing stirring. [1] P. R. Kramer and S. R. Keating. Chin. Ann. Math., 30B(5):631-644, 2009. [2] A. J. Majda and P. R. Kramer. Physics Reports, 314:237-574, 1999. [3] J.-L. Thiffeault. Nonlinearity, 25(2):R1-R44, 2012.   

Learning equations of transport phenomena and fluid dynamics from data

Partial Differential Equations (PDEs) are models that govern transport phenomena and fluid dynamics. Recent advances in data-driven techniques as well as the availability of vast volumes of data from experiments and simulations brought about attention in using machine learning (ML) methods to uncover the structure of equations. In this study, an ML method is proposed to identify particular PDEs including convection-diffusion equations and wave equations. First, useful features are extracted from spatiotemporal data samples to represent the physical behaviors of mathematical terms. Diffusion, Convection, local time-dependent change, and system energy are some potential features. Second, a data-driven model exploits these features to identify the terms that are present in the equation. Incorporating prior knowledge leads to the robustness of the extracted features compared to detected features by Convolutional Neural Networks (CNNs) provided limited amounts of data. The framework presented in this work is efficient as it does not require numerical differentiation or time-consuming network training. Furthermore, the trained model can identify 2D PDEs with time derivatives of different orders, and discover equations out of the training domain.   

Solution to the Nemchinov-Dyson Problem in $2$-D Axial Geometry

The purpose of this work is to examine the solutions to the $2$-D inviscid compressible flow (Euler) equations in axial geometry subject to an ideal gas equation of state (EOS) constrained by the Nemchinov-Dyson assumption on the included velocity field. Assuming a separable solution for the flow velocities $u_r$ and $u_z$ which is defined as a linear spatial component and an arbitrary time function $R_r(t)$ and $R_z(t)$, respectively, we find we find several solution sets for density ($\rho$) , pressure ($P$), and specific internal energy (SIE) ($I$) that are constrained by two ordinary differential equations and arbitrary spatial dependence. These spatial functions are defined as $\Pi\left(\xi,\eta\right)$, $\beta\left(\xi,\eta\right)$, $\Upsilon\left(\xi,\eta\right)$ for $\rho$, $P$, and $I$, respectively, for similarity variables $\xi=r/R_r(t)$ and $\eta=z/R_z(t)$. Using various physically-relevant initial conditions, we find 11 unique numerical solutions to the functional form of $R_r(t)$ and $R_z(t)$. Using different initial density profiles, with assumptions connecting back to uniform thermodynamic properties, we derive specific unique spatial functions for $\rho$, $P$, and $I$. Finally, we show the overall solutions to $\rho$, $P$, and $I$.