Vortex extrusion (VE) is a severe plastic deformation technique which is based on the synergies between high strain accumulation and high hydrostatic pressure. Such a high amount of pressure places a mandate to seek the method for investigation of the load under processing conditions. For this, kinematically admissible velocity field and upper bound terms based on Bezier formulation are developed in order to investigate relative pressure in the VE process. Effects of reduction in area, relative length, twist angle, and friction factor in power dissipation terms are systematically analyzed. It is demonstrated that the increase in twist angle, area reduction and friction factor in the VE process increases the relative pressure, which the rates of these increase vary with twist angle. Moreover, the effect of the relative length is different in various frictional conditions. Results of conventional extrusion (CE) are in good agreement with those found by Avitzur for the effect of slug length and friction factor on the relative extrusion stress.

In this study, the effect of geometrical parameters of rectangular Vortex extrusion die, including twist angle (𝜑 ), reduction in area (𝑅 𝐴 ), and twist zone length (𝐿 ), on deformation inhomogeneity is investigated using the finite element analysis and response surface methodology. Analysis of variance (ANOVA) was used to determine the primary parameters and accuracy of the mathematical model obtained from response surface methodology results. The results showed that the suggested mathematical model predicts the strain inhomogeneity with high accuracy. Additionally, it revealed that the input parameters of 𝜑 , 𝑅 𝐴 , 𝐿 , the interaction between 𝜑 and 𝑅 𝐴 (𝜑 × 𝑅 𝐴 ), and interaction between 𝑅 𝐴 and 𝐿 (𝑅 𝐴 × 𝐿 ) are the main significant factors affecting the inhomogeneity. Perturbation plots and 3D surface diagrams were used to check the results of ANOVA, the sign, and the magnitude of the coefficient of the suggested mathematical model.

The effect of geometrical parameters involved in Vortex extrusion (VE) die design, on AA1050 aluminium alloy processed by VE were investigated using finite element analysis (FEA) and response surface methodology (RSM). For this, VE die length (L), reduction in area (RA), twist angle (𝜑 ), and position of control points in Beziers' formulation (C1) were considered as input parameters and strain inhomogeneity was considered as a response. Both standard deviation (S. D) and inhomogeneity index (Ci) were used to quantify the strain inhomogeneity from FEA results. Analysis of variance (ANOVA) was used to determine the significant parameters and to mathematically model the strain inhomogeneity. It was concluded that standard deviation (S. D) is not a good choice for examining the strain inhomogeneity distribution in VE technique. ANOVA results showed that 𝜑 , RA, and interaction between 𝜑 and RA are the most significant parameters affecting the strain inhomogeneity.

Physical modeling of extrusion process has been studied to obtain qualitative and quantitative information. To observe the flow pattern during the process the plasticine specimen was made with layers of different colors (dark and light). The strain distribution was obtained by measuring the thickness of the plasticine layers and considering the axi-symmetric formulations. The stress distribution and subsequently the extrusion load were computed according to the strain distribution. To model the real frictional effect several ring compression tests have been performed. In order to verify the validity of the modeling data, a similarity study between plasticine and Al 2024-T4 was made. The load obtained by this technique was compared with the results of ANSYS modeling and slab method analysis.

To investigate the dynamics of droplet-Vortex interactions in particle-laden Karman Vortex street flows, the simulations were carried out by using Euler-Lagrange approach, which was validated by the available experiments and numerical results. Then, the particle dispersion and the dimensionless frequency (Strouhal number) of the wake flow were analyzed to evaluate the particle-Vortex interactions. The particle dispersion was statistically analyzed from both time and space dimensions and the different instantaneous dispersion patterns were explained by the relative slip velocity. Two independent scaling parameters, Stokes number StL and particle-fluid mass loading ratio Φ were revealed, and the particle mean square displacement and the Strouhal number were modelled by using these two scaling parameters, respectively. Finally, the characteristic lengths of the particle-laden wake flow were researched, and the Strouhal number physical model was developed based on the oscillating fishtail model. The results indicated that, firstly, StL and Φ, which constitute a dominant scaling group, can characterize the dynamics of droplet-Vortex interactions in wake flow. Particles gradually separate from the Vortex with the increase of StL due to the centrifugal effect, and the Vortex intensity and regularity get worse with the increase of Φ, which further disperses the droplets for their momentum exchange with irregular Vortex structures. Secondly, the length of the formation region and the width of the free shear layer diffuse are the two simultaneous characteristic lengths of the Strouhal number in oscillating wake. The proposed Strouhal number model gives a physical basis for the frequency determination, and the predicted errors are within ±1. 5% error bands with mean absolute percentage error of 0. 67%.