Physics-informed Neural Networks approach to solve the Blasius function

Physics-informed neural networks (PINNs) play a huge role where fewer data and the system’s conventional physics are known to get the solutions to a differential equation. NN can approximate the error to ground truth function, generalize to unseen data and train the model. The contribution of PINNs is to obtain a neural network that knows about the physics hidden behind the equation and thus efficiently solves differential equations. The earlier numerical methods for solving them are finite difference, finite elements, spectral elements, and finite volume. When working with PINNs, there is data efficiency from a machine learning perspective since it is regularized heavily with physics. The required derivatives can be calculated using automatic differentiation at the end of the network. It is possible to train a nonlinear neural network using this approach and its layers of hidden nodes [4]. It then finds a loss function corresponding to the differential equation and boundary conditions.

The boundary layer of a flowing fluid is a thin layer near a solid surface, and the flow near the solid surface is known as the boundary layer flows. Ludwig Prandtl is credited for developing the boundary layer theory. In 1904, he published a paper titled ”On the Motion of a Fluid with Very Small Viscosity” [5]. He laid out the mathematical foundation for flows and condensed the two-dimensional Navier-Stokes equations (NSE) into the boundary layer equations. This publication made understanding fluid motion physics possible, which is regarded as the beginning of contemporary fluid mechanics [6]. The primary disadvantage of using a CFD solver for turbomachinery optimization is the amount of time needed to complete the numerous computationally demanding simulations [7].

The introduction of the similarity variables and the transformation of the PDEs into nonlinear ODEs in one coordinate allowed the successful resolution of the problem. Still, specific methods were needed to handle the unbounded boundary conditions. The Blasius benchmark problem has been resolved [8] using the trial function method put out by Lagaris [9] or a hybrid approach [10]. The solution function is seen to have a singularity on the negative real axis at approximately -5.69. The Blasius equation is a single equation that solely models the viscous boundary layer and is one of CFD’s fundamental models.

In this paper, we propose solving the Blasius equation using PINNs and comparing the solution we obtain with the best-known solutions available in the literature. We shall also extend our solution to the negative real axis to locate the singularity existing for the function. In the next section of the paper, we shall present a brief literature review on the Blasius equation and PINNs. In Section III, we present the methodology that has been adopted to solve the Blasius Equation. Section IV discusses the results and analysis of the proposed algorithm. The conclusions are discussed in the last section.

Neural networks can be used to attain solutions to ODEs and PDEs by reducing them to an optimization problem instead of numerically solving the equations. The application of NNs in fluid mechanics began in the 1990s. Raissi and his team [11, 12] developed a technique called Physics informed neural network (PINN) where the loss function defined in the corresponding NN is extracted from the physics behind the PDE and related equations. In earlier years, Computational fluid dynamics (CFD) had been a great relief in numerically solving the compressible and incompressible Naïve Stokes equations. Minimizing the loss function is challenging in PINNs, as it can be highly complex. Nevertheless, PINNs have been proven to be more accurate than conventional CFD techniques with lesser computations [13].

Although numerical discretization of the Navier-Stokes equations (NSE) has made significant progress in simulating flow problems over the past 50 years, the present algorithms cannot solve governed by high-parametrized NSE. Additionally, it is expensive to solve inverse flow problems due to their complexity, expensive formulations, and need for new algorithms. PINNs are expanded to fractional PINNs (fPINNs) that explore their convergence methodically to solve space-time fractional advection-diffusion equations (fractional ADEs) [14]. A hybrid method for building the residual in the loss function that utilizes both automated differentiation for the integer-order operators and numerical discretization for the fractional operators is a novel component of the fPINNs.

Wide varieties of PINNs have been found in the literature in recent times. can-PINNs [15] link derivative terms with nearby support points, which generally apply to Taylor series expansion-based numerical systems. Apart from demonstrating good dispersion and dissipation characteristics, they are highly trainable and require four to sixteen times fewer collocation points than original PINNs. Auxiliary PINNs (A-PINN) is a technique for solving forward and inverse problems of non-linear integrodifferential equations [16]. ViscoelasticNet is another PINN framework for stress discovery and model selection [17]. PINNs can also be used to solve Reynolds-averaged Navier-Stokes equations [18], full waveform seismic inversions in 2D acoustic media, and wave propagation as it seamlessly handles boundary conditions and physical constraints [19]. In addition to addressing ill-posed problems beyond the scope of conventional computing techniques, PINNs can also close the discrepancy between computational and experimental heat transfer [20].

Blasius equation is a third order non-linear ordinary differential equation of the form $f^{\prime\prime\prime}+\frac{1}{2}ff^{\prime\prime}=0$ with the boundary conditions $f(0)=0$, $f^{\prime}(0)=0$, $f^{\prime}(\infty)=1$. It governs the boundary layer flow over a semi-infinite flat plate. Suppose that the $u-$velocity, the velocity parallel to the surface, is much greater than the $v-$velocity, perpendicular to the surface, and the changes in the perpendicular direction to the surface are much greater than changes parallel to the surface. The boundary layer equations consists of conservation of mass (II.1), conservation of x-momentum (II.2), and conservation of y-momentum. In a flat plate boundary layer, the pressure gradient term appearing in the x-momentum equation becomes zero (II.2). This leads to the hydrodynamic solution for the flat plate boundary layer in a laminar flow called the Blasius solution.

Blasius’ analysis focus on the laminar boundary layer forming on a flat plate. The main aspect of the work is the transformation of the PDE for a flat plate boundary layer with zero pressure gradient into a single ordinary differential equation(ODE) by considering the velocity components that satisfy equation II.1.

The stream function $\psi=U_{\infty}\sqrt{\frac{\nu x}{U_{\infty}}}f(\eta)$, is directly proportional to the function $f(\eta)$ called the Blasius function. Here, $U_{\infty}$ is the free stream velocity, and $\eta$ is a transformed coordinate called the similarity parameter. Here, the velocity components are proportional to the first derivative of $f(\eta)$, and the second and third derivatives of $f$ are proportional to the first and second derivatives of velocity. Substituting these relations into the momentum equation (II.2), the final form of the Blasius boundary layer equation $f^{\prime\prime\prime}+\frac{1}{2}ff^{\prime\prime}=0$, for a flat plate can be obtained. The first and second derivatives of $f(\eta)$ are given by $f^{\prime}=\frac{u}{U_{\infty}}$ (Non-dimensional velocity profile) and $f^{\prime\prime}=\frac{1}{U_{\infty}}\sqrt{\frac{\nu x}{U_{\infty}}}\frac{\partial u}{\partial y}$ (quantity related to shear stress). The boundary conditions are set considering the laminar flow on a flat plate, the no-slip condition, and free-stream velocity outside the boundary layer. Hence we have

Since the value of $f^{\prime\prime}$ at $\eta=0$ is unknown, it cannot be considered an initial value problem.

A power series solution to the boundary layer equation of flow across a flat plate was proposed by Blasius [21]. Schmidt and Beckmann [22] and Ostrach [23] conducted the most important work on the topic, conducting theoretical and experimental research on the free convection flow of air around a vertical flat plate under the influence of gravity. Boyd solved the equation using an analytical series solution technique [24]. Nowadays, with the availability of computers, we can obtain a numerical solution to this equation and calculate it with a very high degree of accuracy. To solve the Blasius equation numerically, a possible method is to use the shooting algorithm to find what value would satisfy the boundary condition at $\eta=0$. The first step is to guess a value at the wall. Then solving the ODE along the non-dimensional coordinate until the first derivative of $f$ stops changing. This checks if the first derivative satisfies the given boundary condition; instead, adjust the guessed value to decrease or increase if the first derivative is higher or lower than one. The algorithm is repeated until the boundary condition is finally satisfied to reach the final solution. Howarth [25] found the solution using a numerical method that accurately predicted the value of $f^{\prime\prime}(0)$ to be 0.332.

This differential equation is a direct representation of the velocity profile inside the boundary layer. Once the solution is obtained, the velocity components can be calculated using these relations. Hence the boundary layer thickness can be calculated located at the point where the velocity is 99$\%$ of the free stream velocity. It was found that this occurs approximately at $\eta$ equal to 5. The displacement and momentum thickness can also be estimated using the solution. The wall shear stress based on the velocity gradient at the wall can be estimated, and the friction and drag coefficients can be calculated. Note that the expression is only accounting for one side of the plate. Comparing the exact solution with the approximation solution from the integral analysis is interesting. The relations are close to each other. Indeed the integral analysis is within 10$\%$ of the exact solution; unlike Blasius’ solution, these values were obtained without complicated math. The Blasius solution provides a self-similar solution meaning that the solution is the same if the independent and dependent variables of the governing equations are appropriately scaled. This can be seen in comparing experimental results with the Blasius solution. Different Reynolds numbers ($Re$) provide the same profile when the variables on the two axes are appropriately scaled.

The solution to the Blasius equation can be found in [26, 27], where the accurate benchmark results of the Blasius boundary layer problem using a leaping Taylors series that converges for all real values. There have been different methods to solve the Blasius equation, such as the Topher transformation, which is executed using inverse transformation, [28, 29, 30]. Runge-Kutta, incorporated with the shooting method, finds the solution numerically, while Adomian Decomposition Method [31] finds the solution analytically.