Combined proper orthogonal decompositions of orthogonal subspaces

We present a method for combining proper orthogonal decomposition (POD) bases optimized with respect to different norms into a single complete basis. We produce a basis combining decompositions optimized with respect to turbulent kinetic energy (TKE) and dissipation rate. The method consists of projecting a data set into the subspace spanned by the lowest several TKE optimized POD modes, followed by decomposing the complementary component of the data set using dissipation optimized POD velocity modes. The method can be fine-tuned by varying the number of TKE optimized modes, and may be generalized to accommodate any combination of decompositions. We show that the combined basis reduces the degree of non-orthogonality compared to dissipation optimized velocity modes. The convergence rate of the combined modal reconstruction of the TKE production is shown to exceed that of the energy and dissipation based decompositions. This is achieved by utilizing the different spatial focuses of TKE and dissipation optimized decompositions.Comment: 9 pages, 3 figure

Personalized modeling for real-time pressure ulcer prevention in sitting posture

, Ischial pressure ulcer is an important risk for every paraplegic person and a major public health issue. Pressure ulcers appear following excessive compression of buttock's soft tissues by bony structures, and particularly in ischial and sacral bones. Current prevention techniques are mainly based on daily skin inspection to spot red patches or injuries. Nevertheless, most pressure ulcers occur internally and are difficult to detect early. Estimating internal strains within soft tissues could help to evaluate the risk of pressure ulcer. A subject-specific biomechanical model could be used to assess internal strains from measured skin surface pressures. However, a realistic 3D non-linear Finite Element buttock model, with different layers of tissue materials for skin, fat and muscles, requires somewhere between minutes and hours to compute, therefore forbidding its use in a real-time daily prevention context. In this article, we propose to optimize these computations by using a reduced order modeling technique (ROM) based on proper orthogonal decompositions of the pressure and strain fields coupled with a machine learning method. ROM allows strains to be evaluated inside the model interactively (i.e. in less than a second) for any pressure field measured below the buttocks. In our case, with only 19 modes of variation of pressure patterns, an error divergence of one percent is observed compared to the full scale simulation for evaluating the strain field. This reduced model could therefore be the first step towards interactive pressure ulcer prevention in a daily setup. Highlights-Buttocks biomechanical modelling,-Reduced order model,-Daily pressure ulcer prevention

A tale of two airfoils: resolvent-based modelling of an oscillator vs. an amplifier from an experimental mean

The flows around a NACA 0018 airfoil at a Reynolds number of 10250 and angles of attack of alpha = 0 (A0) and alpha = 10 (A10) are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions (SPOD) of the velocity fluctuations and nonlinear forcing find low-rank behaviour at the shedding frequency and its higher harmonics for the A0 case. In the A10 case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies low-rank behaviour only in the vicinity of the shedding frequency for A0 and none of its harmonics. The resolvent operator for the A10 case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by SPOD. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow due to the fact that experimental data are missing near the leading edge. The A0 case is classified as an oscillator where the flow is organised around an intrinsic instability while the A10 case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from SPOD when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes.Comment: 32 pages, 23 figure

Numerical investigation of sheet cavitation over a 3-D venturi configuration

Sheet cavitation appears in many hydraulic applications and can lead to technical issues. Numerical simulation is a pertinent way to study the phenomenon. A numerical tool based on 1-fluid compressible RANS equations with a cavitation model is used to compute a flow within a 3-D venturi geometry with a 4° divergent angle. In the present work, a detailed study of this cavitating flow, which presents a quasi-stable vapour pocket, is carried out using tools such as Power Spectral Densities or Spectral Proper Orthogonal Decompositions. An oblique oscillation of the cavity is then identified and discussed

Variational Principles for Stochastic Fluid Dynamics

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The Legendre transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their It\^o representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent It\^o representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to It\^o transformation. This term is a geometric generalisation of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics (SGFD); namely, the Euler-Boussinesq and quasigeostropic approximations.Comment: 19 pages, no figures, 2nd version. To appear in Proc Roy Soc A. Comments to author are still welcome

Structural dissimilarity of large-scale structures in turbulent flows over wavy walls

Turbulent flows over a wavy wall are investigated in channels with a wavy bottom and a flat top with different channel heights. Flow structures are determined from proper orthogonal decompositions of velocity fields measured with particle image velocimetry. Three different channel heights are considered, which are characterized by blockage ratios β (half channel height to wave amplitude ratio) 3.3, 6.7, and 10. Measurements are evaluated at comparable Reynolds numbers (Re) around 10 000. Structural similarity of large-scale structures, which is valid at β = 6.7 and 10, no longer holds at β = 3.3. Furthermore, characteristic regions of flows over wavy walls exhibit different locations in the case of the smallest channel height

On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition

This paper focuses on the efficient solution of models defined in high dimensional spaces. Those models involve numerous numerical challenges because of their associated curse of dimensionality. It is well known that in mesh-based discrete models the complexity (degrees of freedom) scales exponentially with the dimension of the space. Many models encountered in computational science and engineering involve numerous dimensions called configurational coordinates. Some examples are the models encoun- tered in biology making use of the chemical master equation, quantum chemistry involving the solution of the Schrödinger or Dirac equations, kinetic theory descriptions of complex systems based on the solution of the so-called Fokker–Planck equation, stochastic models in which the random variables are included as new coordinates, financial mathematics, etc. This paper revisits the curse of dimensionality and proposes an efficient strategy for circumventing such challenging issue. This strategy, based on the use of a Proper Generalized Decomposition, is specially well suited to treat the multidimensional parametric equations