Numerical Analysis of Transonic Flow around Cones

A new solver is presented for transonic flow around cone-cylinder, axisymmetric bodies. Ground experiments almost always suffer from uncertainty due to operating in the presence of high levels of facility noise. Besides, experimental measurements of these mechanisms are not available at high-speed flows. Direct Numerical Simulations have made it possible to compute details of the transonic mechanisms but still a significant challenge due to the cost. This study aims to present a new solver to model transonic flows. To assess the new solver, the surface Mach number and the drag coefficient are investigated as the freestream Mach number varies. The results are in excellent agreement with experimental data, indicating the new model is capable of accurately predicting the aerodynamics coefficients at transonic flow regimes.


Introduction
Transonic flow past certain two-dimensional bodies has been studied extensively, and therefore, the phenomena are well understood. Some of the earliest theoretical studies are done by Cole [1], Guderley [2], and Vincenti [3], which applied to two-dimensional wedge airfoils. The theory and experimental results conducted by Bryson [4] and Griffith [5] agreed well. Two-dimensional and axially symmetric bodies are of considerable theoretical and practical interest to study since these two cases are simplified cases of the problem around complex and arbitrary geometries.
The study of transonic flow around axisymmetrical bodies is not as complete as two-dimensional bodies. The similarity physicals of axially symmetric transonic flow studied by Von-Karman [6] and Oswatitsch [7] discuss general tran-How to cite this paper: Zangeneh, R.
(2020) Numerical Analysis of Transonic sonic flow past finite cones. The first theoretical results for supersonic flow past a cone were presented in 1929 by Busemann [8]. The Busemann's solution predicts the smooth shock-free compression from supersonic to subsonic flow for specific combinations of cone angle and freestream Mach number. The conical solution also showed that for a given freestream Mach number and cone angle, the surface Mach number is less than the freestream Mach number right behind the conical shock wave; as the freestream Mach number decreases, the surface Mach number also decreases and finally changes from supersonic to subsonic values. It was concluded that the conical solution for the semi-infinite cone is valid for a finite cone only when the flow is supersonic everywhere. However, when the surface Mach number becomes subsonic, the perturbation due to the corner or the cone's shoulder propagates forward through the subsonic field and therefore interrupts the conicity of the flow. Thereby, the conical solution applies only for large enough freestream Mach numbers so that the freestream Mach number is supersonic. For the first time, Taylor and Maccoll [9] presented the numerical solution of the axisymmetric conical flow around semi-infinite cones and validated their experimental data results. The experiments and theoretical results showed notable discrepancies, especially in the shockwave form when the surface Mach number is subsonic. Yoshihara [10] computed the flow around a cone cylinder at the sonic Mach number, which was verified experimentally. The theoretical solution, however, has not been developed for transonic flow around finite cones. Solomon [11] reported the experiment results on several interesting characteristics of the transonic flow around finite cones. The experiment evaluated the deviation of Mach number from the predicted value of conical theory for the transonic freestream Mach number, which leads to an evaluation of drag coefficients.
In the design of aerodynamic vehicles such as missiles, rockets, space shuttles, etc., various shapes are used to reduce the aerodynamic drag to achieve the best performance. Investigating the different parameters on the flow/shock characteristics, such as the shape of the shock wave near the nose, shock detachment distance, and the local Mach number is of special importance in order to determine the parameters that provide minimum aerodynamic drag since drag reduction is essential for the better performance of the aerodynamic vehicles. The formation of the bow shock in the vicinity of the nose, shock detachment distance, shock layer, flow turning angle, etc. plays a substantial role in modifying the aerodynamic characteristic to achieve better cones' performance.
Despite several studies that have been done on flow past different cones num-

A Hybrid-Energetic Numerical Model
To address the conflict between turbulence modeling and shock capturing schemes, a hybrid algorithm is a natural solution, where a dissipative scheme can capture discontinuities at shocks, and the non-dissipative scheme resolves the small scales at the turbulent region. Here an energy conserving scheme is achieved in a finite volume frame-work through hybridization of convective terms, which is evaluated as the inviscid flux, within the Navier-Stokes equations as follows: ( )

Hybrid-Energetic
Central-nondissipative Shock-capturing The non-dissipative component is based on an improved skew-symmetric formulation developed by Kennedy and Gruber [12] for convection terms. In this respect, for the quadratically nonlinear terms, the following single generalized expression is used: where u is the velocity vector and ϑ denotes a generic transport variable. The should be employed. This form is proved to minimize the aliasing errors that are associated with low-order non-dissipative schemes. Here, a modified form is used to compute the convection terms as: where P and N are cell centroids ( Figure 1). The cubically non-linear terms can be discretized in the same manner. It is noted that using second-order methods are common within applications in complex geometries, even for LES applications [13].
A key role in shock-capturing schemes is played by "shock sensors," that must be defined to confine numerical dissipation in shocked regions effectively and, at the same time, not to effect smooth parts of the flow field [14]. The pertinence of the choice of a sensor based on the pressure gradient, capture shock discontinuities usually found in aerodynamics is discussed by Swanson and Turkle [15]. Ducros et al. [13] developed a new correction to the sensor by multiplying the standard sensor by the local function κ , which is defined as the following: is a small positive real number chosen to prevent numerical divergence in regions where is zero. The sensor is able to evaluate the smoothness if the numerical solution by 0 κ = in the smooth zones, and 1 κ = in the presence of shocks [13]. This function alters between 0 for weakly compressible regions to about 1 in shock regions. Here the following modified shock sensor is used as follows: The modified sensor shows smooth correction, proportional to the degree of local compressibility and is proved to predict the right decay of turbulence kinetic energy in turbulent regions out of the shock [13].

Central-Upwind Scheme for Compressible Flows
The Shock-capturing ∅ component in the hybrid algorithm in Equation (1) where f φ is volumetric fluxes. For compressible flows, however, fluid properties are transported by the propagation of waves in addition to transport by the flow. Therefore, the flux interpolation should be stabilized based on transports in any direction [19]. Since the interpolation is done to a given face only from neighboring cell values, the original form of Kurganov and Tadmor (KT) and Kurganov, Noelle, and Petrova (KNP) methods are used [19]. The interpolation procedure is based on splitting into two directions corresponding to flow outward and inward of the face owner cell. The discretization is as follows: The last term in the above equation is volumetric fluxes associated with the local speed of propagation. It is an extra diffusion term related to the maximum speed of propagation of any discontinuity that may occur at a face. Therefore, the value is interpolated in the f − and f + directions. The diffusive volumetric flux is calculated according to: where f ϕ − and f ϕ + are defined as: Here in the interpolation procedure to switch between low and high-order schemes where r represents the ratio of successive gradients of the interpolated variable.
It can be described according to: where ( ) p ∇Ψ is the full gradient calculated at the owner cell, P, and Then, the f + interpolation of Ψ , for example, is evaluated as: ( ) where ( ) van Leer et al. [20]. The resolution of this semi-discrete central-upwind scheme can be further improved, especially of the contact waves, by adding a correction The "correction" term, f q is, in fact, a built-in anti-diffusion term, can be computed as [21]: where: These terms help to reduce the numerical dissipation present in the original form of the semi-discrete central-upwind scheme. Finally, the Laplacian term on the polyhedral cell is discretized by splitting into orthogonal and non-orthogonal components as following: where г is the diffusion coefficient which is interpolated linearly from the cell center values,

Implementation in OpenFOAM
The new framework is developed using C++ code and linked to the existing den-

Computational Domain
The  Figure 1. Only the axisymmetry model about z-axis was simulated due to symmetry.

M ∞ =
, is slightly less than the detachment Mach number predicted by the exact conical theory [22].

Surface Mach Number Distribution
The distribution of the surface Mach number, on a 20 deg cone versus various values of the freestream Mach number at different stations, x/c, where c is the length of the model, is shown in Figure 5 and for the 25 deg cone in Figure 6.
The values of Surface Mach number close to the tip agrees well with the experimental values (Solomon, 1955 [11]). At the corner of the cone, the Mach number should approach sonic, although the boundary layer can affect that. As the freestream Mach number reaches the sonic condition, the surface Mach number approaches a constant value. The behavior is more apparent for the cone angle of 25 deg. This concept is proven from conical theory and is established for axisymmetric flows. The stationary concept can be generalized to three-dimensional bodies for some selective ranges of near sonic velocity.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.