Turbulent boundary layer flow over regularly and irregularly arranged truncated cone surfaces

Aiming to study the rough-wall turbulent boundary layer structure over differently arranged roughness elements, an experimental study was conducted on flows with regular and random roughness. Varying planform densities of truncated cone roughness elements in a square staggered pattern were investigated. The same planform densities were also investigated in random arrangements. Velocity statistics were measured via two-component laser Doppler velocimetry and stereoscopic particle image velocimetry. Friction velocity, thickness, roughness length and zero-plane displacement, determined from spatially averaged flow statistics, showed only minor differences between the regular and random arrangements at the same density. Recent a priori morphometric and statistical drag prediction methods were evaluated against experimentally determined roughness length. Observed differences between regular and random surface flow parameters were due to the presence of secondary flows which manifest as high-momentum pathways and low-momentum pathways in the streamwise velocity. Contrary to expectation, these secondary flows were present over the random surfaces and not discernible over the regular surfaces. Previously identified streamwise-coherent spanwise roughness heterogeneity does not seem to be present, suggesting that such roughness heterogeneity is not necessary to sustain secondary flows. Evidence suggests that the observed secondary flows were initiated at the front edge of the roughness and sustained over irregular roughness. Due to the secondary flows, local turbulent boundary layer profiles do not scale with local wall shear stress but appear to scale with local turbulent shear stress above the roughness canopy. Additionally, quadrant analysis shows distinct changes in the populations of ejection and sweep events.

Ridge-type roughness: from turbulent channel flow to internal combustion engine

While existing engineering tools enable us to predict how homogeneous surface roughness alters drag and heat transfer of near-wall turbulent flows to a certain extent, these tools cannot be reliably applied for heterogeneous rough surfaces. Nevertheless, heterogeneous roughness is a key feature of many applications. In the present work we focus on spanwise heterogeneous roughness, which is known to introduce large-scale secondary motions that can strongly alter the near-wall turbulent flow. While these secondary motions are mostly investigated in canonical turbulent shear flows, we show that ridge-type roughness—one of the two widely investigated types of spanwise heterogeneous roughness—also induces secondary motions in the turbulent flow inside a combustion engine. This indicates that large scale secondary motions can also be found in technical flows, which neither represent classical turbulent equilibrium boundary layers nor are in a statistically steady state. In addition, as the first step towards improved drag predictions for heterogeneous rough surfaces, the Reynolds number dependency of the friction factor for ridge-type roughness is presented. Graphic abstract

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Fluid Gauge Theory

According to the general gauge principle, Fluid Gauge Theory is presented to cover a wider class of flow fields of a perfect fluid without internal energy dissipation under anisotropic stress field. Thus, the theory of fluid mechanics is extended to cover time dependent rotational flows under anisotropic stress field of a compressible perfect fluid, including turbulent flows. Eulerian fluid mechanics is characterized with isotropic pressure stress fields. The study is motivated from three observations. First one is experimental observations reporting large-scale structures coexisting with turbulent flow fields. This encourages a study of how such structures observed experimentally are possible in turbulent shear flows, Second one is a theoretical and mathematical observation: the ”General solution to Euler’s equation of motion” (found by Kambe in 2013) predicts a new set of four background-fields, existing in the linked 4d-spacetime. Third one is a physical query, ”what symmetry implies the current conservation law ?”. The latter two observations encourage a gauge-theoretic formulation by defining a differential one-form representing the interaction between the fluid-current field jμand a background field aμ.

A Constitutive Equation of Turbulence

Even though applications of direct numerical simulations are on the rise, today the most usual method to solve turbulence problems is still to apply a closure scheme of a defined order. It is not the case that a rising order of a turbulence model is always related to a quality improvement. Even more, a conceptual advantage of applying a lowest order turbulence model is that it represents the analogous method to the procedure of introducing a constitutive equation which has brought success to many other areas of physics. First order turbulence models were developed in the 1920s and today seem to be outdated by newer and more sophisticated mathematical-physical closure schemes. However, with the new knowledge of fractal geometry and fractional dynamics, it is worthwhile to step back and reinvestigate these lowest order models. As a result of this and simultaneously introducing generalizations by multiscale analysis, the first order, nonlinear, nonlocal, and fractional Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing (apparent) constitutive equation of turbulent shear flows.

Permuted proper orthogonal decomposition for analysis of advecting structures

Flow data are often decomposed using proper orthogonal decomposition (POD) of the space–time separated form, $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (t\right )\boldsymbol {\phi }_j\left (\boldsymbol {x}\right )$ , which targets spatially correlated flow structures in an optimal manner. This paper analyses permuted POD (PPOD), which decomposes data as $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (\boldsymbol {n}\right )\boldsymbol {\phi }_j\left (s,t\right )$ , where $\boldsymbol {x}=(s,\boldsymbol {n})$ is a general spatial coordinate system, $s$ is the coordinate along the bulk advection direction and $\boldsymbol {n}=(n_1,n_2)$ are along mutually orthogonal directions normal to the advection characteristic. This separation of variables is associated with a fundamentally different inner product space for which PPOD is optimal and targets correlations in $s,t$ space. This paper presents mathematical features of PPOD, followed by analysis of three experimental datasets from high-Reynolds-number, turbulent shear flows: a wake, a swirling annular jet and a jet in cross-flow. In the wake and swirling jet cases, the leading PPOD and space-only POD modes focus on similar features but differ in convergence rates and fidelity in capturing spatial and temporal information. In contrast, the leading PPOD and space-only POD modes for the jet in cross-flow capture completely different features – advecting shear layer structures and flapping of the jet column, respectively. This example demonstrates how the different inner product spaces, which order the PPOD and space-only POD modes according to different measures of variance, provide unique ‘lenses’ into features of advection-dominated flows, allowing complementary insights.