This paper presents a complete derivation and design of a physics-informed neural network (PINN) applicable to solve initial- and boundary value problems described by linear ordinary differential equations. The objective not to develop a numerical solution procedure which is more accurate and efficient than standard finite element or finite difference based methods, but to give a fully explicit mathematical description of a PINN and to present an application example in the context of hydrodynamic lubrication. It is, however, worth noticing that the PINN developed herein, contrary to FEM and FDM, is a meshless method and that training does not require big data which is typical in machine learning.

In this paper, for the Reynolds-averaged Navier-Stokes equations, a self-closed turbulence model without any adjustable parameter is formulated. The validation rule for self-closed turbulence model is rigorously derived from the Reynolds-averaged Navier-Stokes equation. The rule is not effected by turbulence modelling on the Reynolds stresses.

This paper attempts to clarify an long-standing issue about the number of unknowns in the Reynolds-Averaged Navier-Stokes equations (RANS). This study shows that all perspectives regarding the numbers of unknowns in the RANS stem from the misinterpretation of the Reynolds stress tensor. The current literature consider that the Reynolds stress tensor has six unknown components; however, this study shows that the Reynolds stress tensor actually has only three unknown components, namely the three components of fluctuation velocity. This understanding might shed a light to understand the well-known closure problem of turbulence.

This paper attempts to clarify an issue regarding the lasting unsolved problem of turbulence, namely the closure problem. This study shows that all perspectives regarding the numbers of un- known quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor. The current literatures have a consensus that the Reynolds stress tensor has six unknowns; however, this study shows that the Reynolds stress tensor actually has only three ones, namely the three components of fluctuation velocity. With this new understanding, the closed turbulence equations for incompressible flows are proposed.

This paper shown that turbulence closure problem is not an issue at all. All mistakes in the literature regarding the numbers of unknown quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor, i.e., all literatures have stated that the symmetric Reynolds stress tensor has six unknowns; however, it actually has only three unknowns, i.e., the three components of fluctuation velocity. We shown the integral-differential equations of the Reynolds mean and fluctuation equations have exactly eight equations, which equal to the numbers of quantities in total, namely, three components of mean velocity, three components of fluctuation velocity, one mean pressure and one fluctuation pressure. That is why we claim in this paper, that the Reynolds Navier-Stokes turbulence equations of incompressible flow are closed rather than unclosed. This study may help to solve the puzzle that has eluded scientists and mathematicians for centuries.

A century-old scientific conundrum is solved in this paper. The Prandtl mixing length modelled plane boundary turbulent flow is described by: $\frac{du^+}{d{y^+} }+\kappa^2{y^+} ^2\left( \frac{du^+}{d{y^+} }\right)^2=1$, together with boundary condition $ {y^+} =0:\, u^+=0$. Only an approximate solution to this nonlinear ordinary differential equation (ODE) has been sought so far, however, the exact solution to this ODE has not been obtained. By introducing a transformation, $2\kappa y^+=\sinh \xi$, I successfully find the exact solution of the ODE as follows: $u^+=\frac{1}{\kappa}\ln(2\kappa {y^+} +\sqrt{1+4\kappa^2{y^+} ^2})-\frac{2{y^+} }{1+\sqrt{1+4\kappa^2{y^+} ^2}}$.

The open source CFD code OpenFOAM has emerged as one of the most popular alternative to commercial CFD solvers. The recent version of OpenFOAM supports overlapping grids, so called Overset mesh. In this type of mesh, one or more sub-collection of control volumes (or cells) are allowed to overlap with other set of cells. This allows for great flexibility in modelling Fluid Structure Interaction (FSI) problems. Fluid mesh can be standard structured mesh of a rectangular/cuboidal/or cylindrical domain. The structure mesh can be generated separately by creating multiple layers around a given body. The overset mesh functionality allows for overlapping or immersing this structured mesh inside the fluid mesh and a relevant FSI or moving dynamics problems can be solved. The idea of overset mesh has been around since eighties but its support in OpenFOAM is very recent. Most CFD codes which support overset mesh have either been in-house or commercial CFD codes. OpenFOAM's support for overset is the first major open source code resource available for CFD problems. The aim of this research is to solve a classical benchmark airfoil problem using OpenFOAM overset mesh and compare the numerical results with experimental result. We report here flow simulations results around NACA 0018. The result obtained from overset mesh compares convincingly well with experimental results. Computations have been carried out for Reynolds numbers in the range of 10⁵ with angle of attack ranging from α= 5 degree to α= 30 degree with an interval of 5 degree. Turbulence is incorporated using k - ε turbulence model. This study helps developing confidence in using OpenFOAM overset support for more complicated flows and moving dynamics. This report is complemented with a brief description of finite volume discretization using overset mesh.

Calculation of the pressure field on and around solid bodies exposed to external flow is of paramount importance to a number of engineering applications. However, conventional pressure measurement techniques are inherently linked to problems principally caused by their point-wise and/or intrusive nature. In the present paper, we attempt to calculate the time-averaged two-dimensional pressure field by integrating PIV (Particle Image Velocimetry) velocity measurements into a CFD code and modifying them by the respective correction step of the SIMPLE algorithm. Boundary conditions are applied from the PIV data as a three-layer area of constant velocities, adjacent to the boundaries. A novel characteristic of the approach is the straightforward inclusion of the Reynolds Stresses into the source terms of the momentum equations, calculated directly from the PIV statistics. The methodology is applied to three regions of the symmetry plane parallel to the main boundary layer flow past a surface mounted cube. In spite of findings of deviations from the planar 2D flow assumption, the derived pressure fields and the adjusted velocity fields are found to be reliable, while the intrinsic turbulent nature of the flow is considered without modelling of the Reynolds stresses.

This study revisits the Reynolds-averaged Navier--Stokes equations (RANS) and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds' averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of on the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.

The aim of this study is to introduce a novel method which can separate sand or gravel dominated bed load transport in rivers with mixed-size bed material. In engineering practice, the Shields-Parker diagram could be used for such purposes, however, the method has certain applicability limits, due to the fact that it is based on uniform bed material and provides information rather on river-scale, instead of reach or local scale. When dealing with large rivers with complex hydrodynamics and morphodynamics the bed load transport modes can also indicate strong variation even locally, which requires a more suitable approach to estimate the locally unique behavior of the sediment transport. Here, we suggest that the decision criteria utilizes the shear Reynolds number (Re*). The method was verified against field and laboratory measurement data, both performed at non-uniform bed material compositions. The comparative assessment of the results show that the shear Reynolds number based method operates more reliably than the Shields-Parker diagram and it is expected to predict the sand or gravel transport domination with a < 5% uncertainty. The introduced results can greatly contribute to the improvement of numerical sediment transport modeling as well as to the field implementation of bed load transport measurements.

The turbulent Poiseuille flow between two parallel plates is one of the simplest possible physical situations, and it has been studied intensively. In this paper, we propose a modified Prandtl-van Driest mixing length that satisfies both boundary conditions and wall damping effects. With our new formulations, we numerically solve the problem and, moreover, propose an approximate analytical solution of mean velocity. As applications of our solution, an approximate analytical friction coefficient of turbulent Poiseuille flow is proposed.

The common dandelion uses a bundle of drag-enhancing bristles (the pappus) that enables seed dispersal over formidable distances; however, the scaling laws of pneumatic/aerodynamic drag underpinning pappus-mediated flight remains unresolved. In this paper, we will study the pneumatic/aerodynamic shape of dandelion and the scaling law of resistance, and find that the drag resistance coefficient is proportional to the -2/3 power of the dandelion pappus Reynolds number. As a by-product, the terminal velocity analytical expression of the dandelion seed is also obtained.

This paper proposed an explicit and simple representation of velocity fluctuation and the Reynolds stress tensor in terms of the mean velocity field. The proposed turbulence equations are closed. The proposed formulations reveal that the mean vorticity is the key source of producing turbulence. It is found that there are no velocity fluctuation and turbulence if there were no vorticity. As a natural consequence, the laminar- turbulence transition condition was obtained in a rational way.

The aim of work is to derive an explicit expression for a function of vertical mixing induced by wind-waves. To this end, in the Navier-Stokes equations, a current is decomposed into four constituents: the mean flow, the wave-orbital motion, the wave-induced turbulent and the background turbulent currents. This decomposition allows separating the wave-induced Reynolds stress, Rw, from the background one, Rb. To make a statistical closure for Rw, the Prandtl approach for the background turbulent fluctuations is used that results in an implicit expression for the wave-induced vertical mixing function, Bv. Expression for Bv is specified based on the author’s results for the eddy viscosity found earlier in the frame of the three-layer concept for a wavy air–sea interface, used for modelling wind-drift currents [1]. Finally, the explicit parameterization for Bv(a, u*, z) is found as a linear function in both the wave amplitude at depth z, a(z), and the friction velocity in the air, u*. The linear dependence of function Bv(a) on the wave amplitude provides the enhanced vertical mixing induced by wind–waves in comparison with function Bv(a) having the cubic dependence found in [2], as far as the wind-wave amplitude a(z) decays exponentially with depth.

Based on author's previous work [Sun, B. The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed. Preprints 2018, 2018060461 (doi: 10.20944/preprints201806.0461.v1)], this paper proposed an explicit representation of velocity fluctuation and formulated the Reynolds stress tensor in terms of the mean velocity field. The proposed closed Reynolds Navier-Stokes turbulence formulations reveal that the mean vorticity is the key source of producing turbulence.

Micro-technology has played a substantial role in bioscience, biomedical and biotechnological research due to its core advantages in modern science and engineering. It has created unique development in various sectors of bio-research and upsurges the efficacy of research at the molecular level in recent years. Microfluidic technology makes it possible to manipulate sample volumes at the micro- and nano-level (called nanofluidics) with terrific control outside in vivo cellular microenvironment, enabling the reduction of discrepancies between in vivo and in vitro environments as well as reducing reaction time and cost. In this review, we discuss various effective integrations of microfluidic technologies into biotechnology and its paradigmatic significance in bio-research, supporting mechanical and chemical in vitro cellular micro-environment. Specific innovations relating to the application of microfluidics to advance microbial life, solitary and co-cultures along with a multiple-type cell culturing, cellular communications, cellular interactions and population dynamics are discussed.

The exact similarity solutions (also called as special exact solutions) of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, no similarity solutions for two dimensional turbulent boundary layers have been reported in the literature. With the help of dimensional analysis and invariance principle, Prandtl mixing length $\ell=\kappa y$ for one dimensional turbulent boundary layer is extended to $\ell(x,y)=\kappa y (1-\frac{y}{\delta})\sqrt{\frac{\nu}{U\delta}} $ for the two dimensional turbulent boundary layers, furthermore, with a similarity transformation, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation $f'''+ ff''+\beta(1-f'^2)+\kappa^2[\eta^2(1-\eta)^2f''|f''|]'=0$. As an application, similarity solutions of the two dimensional turbulent boundary layer on a flat plate at zero incidence have been studied in detail. To solve the ordinary differential equation numerically, a complete Maple code is provided.

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no similarity solutions (special exact solutions) have ever been found. In the light of Blasius' pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. By author's Maple code, we numerically solve the ordinary differential equation and produce some useful quantities.

Based on the insight gained from single jet analysis performed earlier [1], CFD analysis on turbulent jets impinging on one another at an angle was performed. Multiple impingement angles were considered for this study to gain better understanding of the parameters affecting resultant jet growth and velocity distribution. From the study of single jet, it was concluded that the SST k-ω turbulence model was the ideal turbulence model capable of accurately predicting the flow physics of the jets exiting a fully developed pipe at low Reynolds number. Hence, for the study of impinging jets, SST k-ω turbulence model was used to study the velocity and jet growth characteristics. It became evident that the mesh alignment with the velocity vector at exit of the pipe domain plays a crucial role in the accuracy of the results. The parameter used to evaluate this condition was identified as False Diffusion and was observed to affect the TKE parameter significantly. Methods to reduce false diffusion are also discussed in this article. Based on the mesh obtained from the grid sensitivity study, jets impinging at 30, 45 and 60 degrees at Reynolds number of 7500 were numerically analyzed. It was observed that the profile of the resultant jet closely matched with the prediction of elliptical profile predicted by past researchers, [2] and [3]. Also, it was seen that higher jet growth was predicted in case of jets impinging at a higher impingement angle.

The time-resolved flowfield of a spatially oscillating jet emitted by a Sweeping Jet (SWJ) actuator is investigated numerically using three-dimensional Reynolds-averaged Navier-Stokes (3D-URANS) equations. Numerical simulations are performed for a practical range of mass flow rates providing flow conditions ranging from incompressible to subsonic compressible flows. A linear relationship between the mass flow rate and the jet oscillation frequency is found. The results of the numerical model are compared with the experimental data in the literature, and good agreement is found. Additionally, it is observed that the SWJ actuator frequency response switches from one linear mode to another linear mode when the compressibility effects become important.

In this study, the field measurement-based validation of a novel sediment transport calculation method is presented. River sections with complex bed topography and inhomogeneous bed material composition highlight the need for an improved sediment transport calculation method. The complexity of the morphodynamic features can result in the simultaneous appearance of the gravel and finer sand dominated sediment transport (e.g. parallel bed armoring and siltation) at different regions within a shorter river reach. For the improvement purpose of sediment transport calculation in such complex river beds, a novel sediment transport method was elaborated. The base concept of it is the combined use of two already existing empirical sediment transport models. The method was already validated against laboratory measurements. The major goal of this study is the verification of the novel method with a real river case study. The combining of the two sediment transport models is based on the implementation of a recently presented classification method of the locally dominant sediment transport nature (gravel or sand transport dominates). The results are compared with measured bed change maps. The verification clearly refers to the meaningful improvement in the sediment transport calculation by the novel manner in case of spatially varying bed content.

The aim of the studies was to test the hydraulic pressure losses along the length, local pressure losses, absolute and relative roughness values of the pipeline walls made of modern polymer material. The article presents the results of the experimental studies of hydraulic resistance of pipelines from modern polymer materials, the method of theoretical determination of resistance coefficients in the VALTEC ISO 21003 metal polymer pipeline, the classification and method of determining the values of local hydraulic resistance coefficients of T-Junctions are given, the dependence for determining the Darcy λ hydraulic friction coefficient associated with the value of the equivalent hydraulic roughness coefficient of the inner surface of pipes and the Reynolds number. The experimental data of the hydraulic coefficient of friction Darcy λ was compared with the known theoretical relationships of A.D. Altschul, Colbrooke for smooth pipes, Colbrooke and White, an alternative to the Colbrooke-White, Offengenden equation for low-shear pipes and pipes from polymer materials with Reynolds numbers Re>10^4. Dependence of hydraulic coefficient of friction on relative equivalent roughness of pipeline walls and Reynolds number, specific energy loss per one linear meter on total flow rate in system of hydraulically short pipeline is obtained. The empirical dependence of values of coefficients of local resistance ζ=f(Q2/Q3) from a ratio of the divided expenses of Q2/Q3 of the equal-pass T-Junction α=90 the pipeline to a branch up, on pass in the direct T-Junction and division of streams at various internal pipeline diameters of VALTEC.