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A novel method is proposed to identify vortex boundary and center of rotation based on tubular surfaces of constant stagnation pressure and minimum of the stagnation pressure gradient. The method is derived from Crocco's theorem, which ensures that the gradient of stagnation pressure is orthogonal to both the velocity and vorticity vectors. The method is Galilean invariant, requires little processing and is robust. It enables visualization of complex turbulent flows and provides a physically consistent definition of vortex boundaries for quantitative analyses. This vortex boundary is a material surface that is representative of the kinematics of the flow by construction, constitutes a vortex tube, ensures conservation of circulation in the inviscid limit and provides a unique relation to the conservation of momentum equations and vortex loads., Comment: 9 pages, 8 figures

The influence of roughness spacing on boundary layer transition over distributed roughness elements is studied using direct numerical simulation (DNS) and global stability analysis, and compared to isolated roughness elements at the same Reh. Small spanwise spacing ($\lambda_z = 2.5h$) inhibits the formation of counter-rotating vortices (CVP) and as a result, hairpin vortices are not generated and the downstream shear layer is steady. For $\lambda_z = 5h$, the CVP and hairpin vortices are induced by the first row of roughness, perturbing the downstream shear layer and causing transition. The temporal periodicity of the primary hairpin vortices appears to be independent of the streamwise spacing. Distributed roughness leads to a lower critical Reh for instability to occur and a more significant breakdown of the boundary layer compared to isolated roughness. When the streamwise spacing is comparable to the region of flow separation ($\lambda_x = 5h$), the high-momentum fluid barely moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements, and its frequency is close to the shedding frequency of hairpin vortices in the isolated case. For larger streamwise spacing ($\lambda_x = 10h$), two distinct modes are obtained from global stability analysis. The first mode shows varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks formed in the longitudinal grooves are also found to be unstable and interacting with the varicose mode. The second mode is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs into the second row. It extracts most energy from the spanwise shear between the high- and low-speed streaks.

A variational volume-of-fluid (VVOF) methodology is devised for evolving interfaces under curvature-dependent speed. The interface is reconstructed geometrically using the analytic relations of Scardovelli and Zaleski [1] and the advection of the volume fraction is performed using the algorithm of Weymouth and Yue (WY) [2] with a technique to incorporate a volume conservation constraint. The proposed approach has the advantage of simple implementation and straightforward extension to more complex systems. Canonical curves and surfaces traditionally investigated by the level set (LS) method are tested with the VVOF approach and results are compared with existing work in LS.

This paper presents a methodology to construct a divergence-free polynomial basis of an arbitrary degree in a simplex (triangles in 2D and tetrahedra in 3D) of arbitrary dimension. It allows for fast computation of all numerical solutions from degree zero to a specified degree \textit{k} for certain PDEs. The generated divergence-free basis is orthonormal, hierarchical, and robust in finite-precision arithmetic. At the core is an Arnoldi-based procedure. It constructs an orthonormal and hierarchical basis for multi-dimensional polynomials of degree less than or equal to \textit{k}. The divergence-free basis is generated by combining these polynomial basis functions. An efficient implementation of the hybridized BDM mixed method is developed using these basis functions. Hierarchy allows for incremental construction of the global matrix and the global vector for all degrees (zero to \textit{k}) using the local problem solution computed just for degree \textit{k}. Orthonormality and divergence-free properties simplify the local problem. PDEs considered are Helmholtz, Laplace, and Poisson problems in smooth domains and in a corner domain. These advantages extend to other PDEs such as incompressible Stokes, incompressible Navier-Stokes, and Maxwell equations.

Global stability analysis and direct numerical simulation (DNS) are performed to study boundary layer flows with an isolated roughness element. Wall-attached cuboids with aspect ratios $\eta=1$ and $\eta=0.5$ are investigated for fixed ratio of roughness height to displacement boundary layer thickness $h/\delta^*=2.86$. Global stability analysis is able to capture the frequency of the primary vortical structures. For $\eta=1$, only varicose instability is seen. For the thinner roughness element ($\eta=0.5$), the varicose instability dominates the sinuous instability, and the sinuous instability becomes more pronounced as $Re_h$ increases, due to increased spanwise shear in the near-wake region. The unstable modes mainly extract energy from the central streak, although the lateral streaks also contribute. The DNS results show that different instability features lead to different behavior and development of vortical structures in the nonlinear transition process. For $\eta=1$, the varicose mode is associated with the shedding of hairpin vortices. As $Re_h$ increases, the breakdown of hairpin vortices occurs closer to the roughness and sinuous breakdown behavior promoting transition to turbulence is seen in the farther wake. A fully-developed turbulent flow is established in both the inner and outer layers farther downstream when $Re_h$ is sufficiently high. For $\eta=0.5$, the sinuous wiggling of hairpin vortices is prominent at higher $Re_h$, leading to stronger interactions in the near wake, as a result of combined varicose and sinuous instabilities. A sinuous mode captured by dynamic mode decomposition (DMD) analysis, and associated with the `wiggling' of streaks persists far downstream.

The direct measurement of wall shear stress in turbulent boundary layers (TBL) is challenging, therefore requiring it to be indirectly determined from mean profile measurements. Most popular methods assume the mean streamwise velocity to satisfy either a logarithmic law in the inner layer or a composite velocity profile with many tuned constants for the entire TBL, both of which require reliable data from the inner layer. The presence of roughness and pressure gradient brings additional complications where most existing methods either fail or require significant modification. A novel method is proposed to determine the wall shear stress in zero pressure gradient TBL from measured mean profiles, without requiring near-wall data. The method is based on the stress model of Kumar and Mahesh [Phys. Rev. Fluids 6, 024603 (2021)], who developed accurate models for mean stress and wall-normal velocity in zero pressure gradient TBL. The proposed method requires a single point measurement of mean streamwise velocity and mean shear stress in the outer layer, preferably between $20$ to $50 \%$ of the TBL, and an estimate of boundary layer thickness and shape factor. The method can handle wall roughness without modification and is shown to predict friction velocities to within $3 \%$ over a range of Reynolds number for both smooth and rough wall zero pressure gradient TBL. In order to include the pressure gradients effects, the work of Kumar and Mahesh [Phys. Rev. Fluids 6, 024603 (2021)] is revisited to derive a novel model for both mean stress and wall-normal velocity in pressure gradient TBL, which is then used to formulate a method to obtain the wall shear stress from the profile data. Overall, the proposed method is shown to be robust and accurate for a variety of pressure gradient TBL., Comment: 19 pages, 22 figures, accepted to Experiments in Fluids

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid-structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectra of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the net displacement source reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be $\approx 0.75\delta$. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the location and width of the contribution depend on inner and outer units, respectively.

The sources of wall-pressure fluctuations in turbulent channel flow are studied using a novel framework. The wall-pressure power spectral density (PSD) $(\phi_{pp}(\omega))$ is expressed as an integrated contribution from all wall-parallel plane pairs, $\phi_{pp}(\omega)=\int_{-\delta}^{+\delta}\int_{-\delta}^{+\delta}\Gamma(r,s,\omega)\,\mathrm{dr}\,\mathrm{ds}$, using the Green's function. Here, $\Gamma(r,s,\omega)$ is termed the net source cross spectral density (CSD) between two wall-parallel planes, $y=r$ and $y=s$ and $\delta$ is the half channel height. Direct Numerical Simulation (DNS) data at friction Reynolds number of $180$ and $400$ are used to compute $\Gamma(r,s,\omega)$. Analysis of the net source CSD, $\Gamma(r,s,\omega)$ reveals that the location of dominant sources responsible for the premultiplied peak in the power spectra at $\omega^+\approx 0.35$ and the wavenumber spectra at $\lambda^+\approx 200$ is in the buffer layer at $y^+\approx 16.5$ and $18.4$ for $Re_{\tau}=180$ and $400$, respectively. The contribution from a wall-parallel plane (located at distance $y^+$ from the wall) to wall-pressure PSD is log-normal in $y^+$ for $\omega^+>0.35$. A dominant inner-overlap region interaction of the sources is observed at low frequencies. Further, the decorrelated features of the wall-pressure fluctuation sources are analyzed using spectral Proper Orthogonal Decomposition (POD). We require the modes to be orthogonal in an inner product with a symmetric positive definite kernel. Spectral POD supports the case that the net source is composed of two decorrelated components - active (dominant mode) and inactive (remaining modes). The structure represented by the dominant POD mode at the premultiplied wall-pressure PSD peak inclines in the downstream direction. At the low-frequency linear PSD peak, the dominant mode resembles a large scale vertical pattern.

Cavitating flow over a circular cylinder is investigated over a range of cavitation numbers for both laminar and turbulent regimes. We observe non-cavitating, cyclic and transitional cavitation regimes with reduction in freestream $\sigma$. The cavitation inside the Karman vortices in the cyclic regime, is significantly altered by the onset of "condensation front" propagation in the transitional regime. At the transition, an order of magnitude jump in shedding Strouhal number is observed as the dominant frequency shifts from periodic vortex shedding in the cyclic regime, to irregular-regular vortex shedding in the transitional regime. In addition, a peak in pressure fluctuations, and a maximum in St versus $\sigma$ based on cavity length are observed at the transition. Shedding characteristics in each regime are discussed using dynamic mode decomposition. It is demonstrated that the condensation fronts observed in the transitional regime are supersonic (referred as "condensation shocks"). In the presence of NCG, multiple condensation shocks in a given cycle are required for complete cavity condensation and detachment, as compared to a single condensation shock when only vapor is present. This is explained by the reduction in pressure ratio across the shock in the presence of NCG effectively reducing its strength. In addition, at $\sigma=0.85$ (near transition from the cyclic to the transitional regime), presence of NCG suppresses the low frequency irregular-regular vortex shedding. Vorticity transport at $Re=3900$, in the transitional regime, indicates that the region of attached cavity is nearly two-dimensional with very low vorticity, affecting Karman shedding in the near wake. In addition, the boundary-layer separation point is found to be strongly dependent on the amounts of vapor and gas in the freestream for both laminar and turbulent regimes., Comment: Under review in Journal of Fluid Mechanics

A robust numerical methodology to predict equilibrium interfaces over arbitrary solid surfaces is developed. The kernel of the proposed method is the distance regularized level set equations (DRLSE) with techniques to incorporate the no-penetration and mass-conservation constraints. In this framework, we avoid reinitialization typically used in traditional level set methods. This allows for a more efficient algorithm since only one advection equation is solved, and avoids numerical error associated with the re-distancing step. A novel surface tension distribution, based on harmonic mean, is prescribed such that the zero level set has the correct the liquid-solid surface tension value. This leads to a more accurate triple contact point location. The method uses second-order central difference schemes which facilitates easy parallel implementation, and is validated by comparing to traditional level set methods for canonical problems. The application of the method, in the context of Gibbs free energy minimization, to obtain liquid-air interfaces is validated against existing analytical solutions. The capability of our current methodology to predict equilibrium shapes over both structured and realistic rough surfaces is demonstrated.

This paper studies multiphase flow within grooved textures exposed to external unsteadiness. We derive analytical expressions for multiphase unsteady Stokes flow driven by oscillating streamwise/spanwise velocity in the presence of periodic grooves. Good agreement is obtained between the analytical solution and direct numerical simulations performed using the volume of fluid method. The effects of oscillation frequency, Reynolds number, and the multiphase interface location on the transfer function between the input signal external to the groove and output near the interface, are examined. The effective slip length and the shear stress over the grooved plane are described., Comment: Submitted to the Journal of Fluid Mechanics

A novel technique based on the Full Orthogonalization Arnoldi (FOA) is proposed to perform Dynamic Mode Decomposition (DMD) for a sequence of snapshots. A modification to FOA is presented for situations where the matrix $A$ is unknown, but the set of vectors $\{A^{i-1}v_1\}_{i=1}^{N-1}$ are known. The modified FOA is the kernel for the proposed projected DMD algorithm termed, FOA based DMD. The proposed algorithm to compute DMD modes and eigenvalues i) does not require Singular Value Decomposition (SVD) for snapshot matrices $X$ with $\kappa_2(X) \ll 1/\epsilon_m$, where $\kappa_2(X)$ is the 2-norm condition number of the snapshot matrix and $\epsilon_m$ is the relative round-off error or machine epsilon, ii) has an optional rank truncation step motivated by round off error analysis for snapshot matrices $X$ with $\kappa_2(X) \approx 1/\epsilon_m$, iii) requires only one snapshot at a time, thus making it a 'streaming' method even with the optional rank truncation step, iv) consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected DMD methods and v) lends itself to easy parallelism as the main computational kernel involves only vector additions, dot products and matrix vector products. The new technique is therefore well-suited for DMD of large datasets on parallel computing platforms. We show both theoretically and using numerical examples that for FOA based DMD without rank truncation, the finite precision error in the computed projection of the linear mapping is $O(\epsilon_m\kappa_2(X))$. The proposed method is also compared to existing projected DMD methods for computational cost, memory consumption and relative round off error. Error indicators are presented that are useful to decide when to stop acquiring new snapshots. The proposed method is applied to several examples of numerical simulations of fluid flow., Comment: accepted in Journal of Computational Physics

Direct numerical simulations (DNS) are performed for two wall-bounded flow configurations: laminar Couette flow at $Re=740$ and turbulent channel flow at $Re_{\tau}=180$, where $\tau$ is the shear stress at the wall. The top wall is smooth and the bottom wall is a realistically rough superhydrophobic surface (SHS), generated from a three-dimensional surface profile measurement. The air-water interface, which is assumed to be flat, is simulated using the volume-of-fluid (VOF) approach. The laminar Couette flow is studied with varying interface heights $h$ to understand the effect on slip and drag reduction ($DR$). The presence of the surface roughness is felt up to $40\%$ of the channel height in the wall-normal direction. A nonlinear dependence of $DR$ on $h$ is observed with three distinct regions. The DNS results are used to obtain a nonlinear curve fit for gas fraction $\phi_g$ as a function of $h$, where $\phi_g$ determines the amount of slip area exposed to the flow. A power law linear regression fit is used to obtain $b_{eff}$ as a function of $\phi_g$. For the turbulent channel flow, statistics of the flow field are compared to that of a smooth wall to understand the effects of roughness and $h$. Two interface heights, $h=h_{rms}$ and $h=h_{max}$ are simulated to study their effect on the behaviour of the flow, where $h_{rms}$ and $h_{max}$ are the rms and maximum peak of the roughness heights respectively. Results show that the presence of trapped air in the cavities significantly alters near wall flow physics. The fully wetted roughness increases the peak value in turbulent intensities, whereas the presence of the interface suppresses them. Overall, there exists a competing effect between the interface and the asperities, where the interface suppresses turbulence whereas the asperities enhance them., Comment: 41 pages, 28 figures, 5 tables. Submitted to the Journal of Fluid Mechanics

Axisymmetric boundary layers are studied using integral analysis of the governing equations for axial flow over a circular cylinder. The analysis includes the effect of pressure gradient and focuses on the effect of transverse curvature on boundary layer parameters such as shape factor ($H$) and skin-friction coefficient ($C_f$), defined as $H = \delta^*/\theta$ and $C_f = \tau_w/(0.5\rho U_e^2)$ respectively, where $\delta^*$ is displacement thickness, $\theta$ is momentum thickness, $\tau_w$ is the shear stress at the wall, $\rho$ is density and $U_e$ is the streamwise velocity at the edge of the boundary layer. Relations are obtained relating the mean wall-normal velocity at the edge of the boundary layer ($V_e$) and $C_f$ to the boundary layer and pressure gradient parameters. The analytical relations reduce to established results for planar boundary layers in the limit of infinite radius of curvature. The relations are used to obtain $C_f$ which shows good agreement with the data reported in the literature. The analytical results are used to discuss different flow regimes of axisymmetric boundary layers in the presence of pressure gradients., Comment: 14 pages, 3 figures, submitted to Journal of Fluid Mechanics