You may never get an answer to your question because you have precluded any viable approach by refusing to utilize curve-fitting of the experimental data. Delft U has an excellent reputation. You are fortunate to have been accepted there. I hope you learn many new things.

Thanks, Dudley. I wish you a happy new year as well. Thanks for your response. I think it would be helpful for the community to fully understand the problem if we can find a semi-analytical equation for the drag coefficient. I'm working on something that may lead to an interesting result. If you'd like, I can share it with you. Thanks again for the well wishes.

Having an analytical expression for the drag, means you have an analytical solution for the velocity field.

There is an analytical solution for flow over a sphere at lo Re number but it is impossibile to produce exact analytical solutions for transitional and turbulent regimes.

At present a good correlation Cd(Re) is available from experinents and simulations.

Why do you refuse to use that?

I have solved similar problems by taking the asymptotic solution for very low Re and the asymptotic solution for very high Re and splicing between them with an analytical function like hyperbolic tangent. Ideas for potential analytical splicing functions can be gathered by looking at the graphics in the CRC Handbook of Mathematical Tables.

Dear Mr. Hasadi,

In your question, you did not provide a complete statement of the problem. Do I understand correctly that you are looking for the dependence of the drag coefficient of an infinite cylinder, installed normal to the uniform flow velocity, on the Reynolds number in an identically homogeneous fluid? If so, then your problem is purely mathematical, having no physical analogue, since the main material parameter of the liquid that is variable density! is excluded from consideration. To analyze similar problems, it suffices to use only the axiomatic concepts of metric space and time. In the two-dimensional approximation, the system of Navier-Stokes equations is solvable; the development of mathematics allows formal searching the problem solution.

However, the adequacy of the obtained solution requires additional evaluation. In real conditions, the density of a liquid is !always! variable. Accounting for the density variability leads to an increase in the rank of the equations of fluid mechanics, which, in the case of small dissipative parameters, can be analyzed by the theory of singular perturbations. At the same time, the class of eigenfunctions that describe the structure and dynamics of flows is significantly expanded. Regular solutions describe waves, singular solutions describe ligaments (interfaces, trickles, fibers, and about 30 more synonyms), which in the experiment correspond to high-gradient fibers and interfaces [1], which significantly change the flow pattern and the drag coefficient. Sets of regular and singular components describe structured wakes and vortices. The effects of density heterogeneity are especially pronounced at low body velocities, when blocking effects are significant, creating an upstream perturbation, as well as sharpening of the gradient, causing the formation of envelopes in the downstream wake. Schlieren visualization of stratified flows behind the sphere is given in [2], past the plate - in [3]. Schlieren images of the flow around a cylinder in various regimes can be found in Russian literature of the end of the last century and the beginning of this century. The properties of ligaments in the linear and weakly nonlinear approximation are analyzed in [4]. The physical nature of the sharpening of the gradients is determined by the inhomogeneity of the distribution of thermodynamic potentials in different parts of the flow and the mechanisms of local transformation of internal energy into other forms on very small scales.

Therefore, the answer of your questions is using the fundamental set of equations for heterogeneous fluids. Thus, in a density stratified liquid, it is of scientific interest to search for the dependence of the drag coefficient on the defining dimensionless parameters that are Reynolds number, internal Froude number Fr=U/ND, and scale ratio C=L/D. Here U is the velocity of a cylinder with a diameter D, N is the buoyancy frequency, L=g/N^2 – stratification scale, inverse logarithmic density gradient. The curve to be found in the approximation of a homogeneous liquid will be a one-dimensional projection of the surface С(sub)x= C(sub)x (Re, Fr, C) in the limit of infinite values of the Froude numbers and the scale ratio, which depends on the choice of the transition pattern. In a more general statement, the fluid density dependence on the other dynamic and thermodynamic parameters may be taken into account.

References.

1. Chashechkin Y.D. Foundations of engineering mathematics applied for fluid flows // Axioms. 2021. V. 10. Iss.4 286. https://doi.org/10.3390/axioms10040286.

2. Chashechkin, Y.D. Discrete and Continuous Symmetries of Stratified Flows Past a Sphere //

Symmetry 2022, 14(6), 1278. DOI:10.3390/sym14061278. https://doi.org/10.3390/sym14061278.

3. Chashechkin Yu. D., Zagumennyi I. V. 2D hydrodynamics of a plate: from creeping flow to transient vortex regimes // Fluids. 2021. V.6(9). 310. https://doi.org/10.3390/fluids6090310

4.Chashechkin, Yuli D. Conventional partial and new complete solutions of the fundamental equations of fluid mechanics in the problem of periodic internal waves with accompanying ligaments generation // Mathematics. 2021. V. 9(6). No. 586. https://doi.org/10.3390/math9060586

Thank you Prof Yuli D. Chashechkin for such an elaborate response. I am not the author (OP) but the description of your response has compelled me to explore the references.

Thank Professor Yuli D. Chashechkin, for your very insightful answer, I am sorry for my extremely late reply, but I was working on a paper that may also shed some information about the problem of flow over bluff bodies in the case of high Reynolds flow regime, we can consider that the flow can be in incompressible flow conditions. The main output of the paper is a model that can predict the drag coefficient of an arbitrarily shaped body. The model can not be considered a fitting model, like many available in the open literature, because it has properties of a closed-form solution, which was the topic of my question.

I published a paper not long ago about a drag coefficient model for the case of the sphere; we used a machine learning method to derive the model from data available in the literature. The main mathematical premise of the model is that it follows the conjecture of Proudman and Pearson (1956), who suggested that the drag coefficient for the case of the sphere will be a function of logarithmic terms of the Reynolds number. We partially proved their conjecture (

Article Do Logarithmic Terms Exist in the Drag Coefficient of a Sing...

)

. Our predictive model extrapolated beyond its training data and managed to predict the drag crisis for the case of the sphere and found the Oseen constant at extremely low Reynolds numbers. Also, In our paper, we derive an analytical solution for the case of forced convective flow over the sphere from data; bear in mind that in our analysis, we did not explicitly use the mathematical form of the Naiver-Stokes equations but only discrete functions, and data points that may represent their solution.

In our recent paper (

Preprint A Generalized Model for Predicting the Drag Coefficient of A...

) We used basically the model that we drive for the case of the sphere, and we found that we only need to adjust one of its coefficients (a1) to predict the drag coefficient of any other geometry. The a1 coefficient is linked to the frictional drag coefficient derived from the boundary layer theory. We tested our hypothesis for different geometries, such as horizontal flat plates, normal flat plates, spheres, and cylinders. For example, we derived the frictional drag obtained by Blasius, which shows that our reduced model is close to the general solution of the Navier- Stokes equations in its incompressible form. Our model can predict the drag coefficient for many geometries for Reynolds numbers up to 10^5. Another new physics that we find about the problem is that the rate of change of the drag coefficient with respect to the Reynolds number is not dependent on the object's geometry. Since our model did not use any training data from other geometries except that of the sphere, we consider it a close mirror version of the solution of the Naiver-Stokes equations. I will be apricated for your input.

Dear Dudley J Benton,

if you are interested, please check our latest work (

Preprint A Generalized Model for Predicting the Drag Coefficient of A...

), we present a model for the prediction of the drag coefficient of arbitrary shape. The model is derived using only the data from the case of the sphere by using a machine learning method. In order to predict the drag coefficient beyond the sphere geometry, you need to change a single coefficient of the model (a1). This coefficient strongly relates to frictional drag derived using the boundary layer theory. We tested our hypothesis for many geometries, such as horizontal flat plates, spheres, cylinders, spheres, and normal flat plates. For example, we derived the fractional drag derived by Blasius for the case of a horizontal flat plate. We are not claiming to get a closed-form solution but a simplified version of it.