A minimization principle for the description of time-dependent modes associated with transient instabilities

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime they can play a crucial role by either altering the system dynamics through the activation of other instabilities, or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighborhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: i) linear systems including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and ii) nonlinear problems including a low-dimensional system with transient instabilities and the vertical jet in crossflow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short time regime), while for longer times the modes capture the expected asymptotic behavior

System-level Modeling of Cooling Networks in All Electric Ships

A Thermal management simulation tool is required to rapidly and accurately evaluates and mitigates the adverse effects of increased heat loads in the initial stages of design in all electric ships. By reducing the dimension of Navier-Stokes and energy equations, we have developed one-dimensional partial differential equations models that simulate time-dependent hydrodynamics and heat transport in a piping network system. Beside the steady-state response, the computational model enables us to predict the transient behavior of the cooling system, when the operating conditions are time-variant. To accurately predict the impact of cooling system on temperature distribution at different ship's locations/components and vice versa, we coupled our computational tool with vemESRDC developed at Florida State University. We verified our implementation with several benchmark problems.United States. National Oceanic and Atmospheric Administration (Grant N000141410166

Uncertainty quantification of film cooling effectiveness in gas turbines

In this study the effect of uncertainty of velocity ratio on jet in crossflow and particual- rly film cooling performance is studied. Direct numerical simulations have been combined with a stochastic collocation approach where the parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) method. Velocity ratio serves as a bifurcation parameter in a jet in a crossflow and the dynamical system is shown to have several bifurcations. As a result of the bifurcations, the target functional is observed to have low-regularity with respect to the paramteric space. In that sense, ME-gPC is particularly effective in discretizing the parametric space. One particular case of a jet in a crossflow is numerically solved with the velocity ratio variations assumed to have a truncated Gaus- sian distribution with mean of 1.5 and the standard variation of approximately 0.5. Five elements are used to discretize the parametric space using ME-gPC method. Within each element general polynomial chaos of order 3 is used. A fast convergence of the polynomial expansion in the parametric space was observed. Time-dependent Navier-Stokes equations are sampled at Gauss-quadrature points using spectral/hp element method implemented in NEKTAR. Overall due to the low-regularity of the response surface, ME-gPC is observed to be a computationally effective strategy to study the effect of uncertainty in a jet in a crossflow when velocity ratio is the random parameter

Analysis and optimization of film cooling effectiveness

In the first part, an optimization strategy is described that combines high-fidelity simu- lations with response surface construction, and is applied to pulsed film cooling for turbine blades. The response surface is constructed for the film cooling effectiveness as a function of duty cycle, in the range of DC between 0.05 and 1, and pulsation frequency St in the range of 0.2-2, using a pseudo-spectral projection method. The jet is fully modulated and the blowing ratio, when the jet is on, is 1.5 in all cases. Overall 73 direct numerical sim- ulations (DNS) using spectral element method were performed to sample the film cooling effectiveness on a Clenshaw-Curtis grid in the design space. It is observed that in the parameter space explored a global optimum exists, and in the present study, the best film cooling effectiveness is found at DC = 0.14 and St = 1.03. In the same range of DC and St, four other local optimums were found. The gradient-based optimization algorithms are argued to be unsuitable for the current problem due to the non-convexity of the objective function. In the second part, the effect of randomness of blowing ratio on film cooling performance is investigated by combining direct numerical simulations with a stochastic collocation ap- proach. The blowing ratio variations are assumed to have a truncated Gaussian distribution with mean of 0.3 and the standard variation of approximately 0.1. The parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) with five elements where general polynomial chaos of order 3 is used in each element. Direct numerical simula- tions were carried out using spectral/hp element method to sample the governing equations in space and time. The probability density function of the film cooling effectiveness was obtained and the standard deviation of the adiabatic film cooling effectiveness on the blade surface was calculated. A maximum standard deviation of 15% was observed in the region within a four-jet-diameter distance downstream of the exit hole. The spatially-averaged adiabatic film cooling effectiveness was 0.23 0.02. The calculation of all the statistical properties were carried out as off-line post-processing. Overall the computational strategy is shown to be very effective with the total computational cost being equivalent to solving twenty independent direct numerical simulations that are performed concurrently. In the third part, an accurate and efficient finite difference method for solving the incompressible Navier-Stokes equations on curvilinear grids is developed. This method combines the favorable features of the staggered grid and semi-staggered grid approaches. A novel symmetric finite difference discretization of the Poisson-Neumann problem on curvilinear grids is also presented. The validity of the method is demonstrated on four benchmark problems. The Taylor-Green vortex problem is solved on a curvilinear grid with highly skewed cells and a second-order convergence in .-norm is observed. The mixed convection in a lid-driven cavity is solved on a highly curvilinear grid and excellent agreement with literature is obtained. The results for flow past a cylinder are compared with the existing experimental data in the literature. As the fourth case, three dimensional time-dependent incompressible flow in a curved tube is solved. The predictions agree well with the measured data, and validate the approach used

Reduced order modeling with time-dependent bases for PDEs with stochastic boundary conditions

Low-rank approximation using time-dependent bases (TDBs) has proven effective for reduced-order modeling of stochastic partial differential equations (SPDEs). In these techniques, the random field is decomposed to a set of deterministic TDBs and time-dependent stochastic coefficients. When applied to SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC must be specified for each of the TDBs. However, determining BCs for TDB is not trivial because: (i) the dimension of the random BCs is different than the rank of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality or orthogonality constraints and specifying BCs for TDB should not violate these constraints in the space-discretized form. In this work, we present a methodology for determining the boundary conditions for TDBs at no additional computational cost beyond that of solving the same SPDE with homogeneous BCs. Our methodology is informed by the fact the TDB evolution equations are the optimality conditions of a variational principle. We leverage the same variational principle to derive an evolution equation for the value of TDB at the boundaries. The presented methodology preserves the orthonormality or orthogonality constraints of TDBs. We present the formulation for both the dynamically bi-orthonormal (DBO) decomposition as well as the dynamically orthogonal (DO) decomposition. We show that the presented methodology can be applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We assess the performance of the presented method for linear advection-diffusion equation, Burgers' equation, and two-dimensional advection-diffusion equation with constant and temperature-dependent conduction coefficient

Parametric Optimization of S-type Cables

We formulated an S-curve parameterization for re-routing a cable path for a given offset displacement. In our approach we assume the cable path to follow a tangent hyperbolic curve. In our formulation, after taking the geometric constraints into account, two free parameters exist. The objective of our design is to maximize the curvature radius. We performed a full search in the two-dimensional design space to find the curvature radius for all possible design configurations. The two-dimensional distribution of curvature radius versus the design parameters is shown in a contour plot that can be readily used for design purposes.United States. National Oceanic and Atmospheric Administration (Grant N000141410166