A numerical analysis of the turbulent flow around NACA 0012 and NACA 4412 hydrofoils is performed to determine the distribution of surface pressure and lift forces for different angles of attack. 𝑈 Contravariant velocity component 𝑢 𝑥 instantaneous velocity component 𝑢𝑃+ Dimensionless velocity in the law of the wall.
Motivation
Objectives of the Research
To calculate the surface pressure distribution on a number of hydrofoils (NACA 0012 and NACA 4412) located far from the free surface and compare the results with available experimental data. iii). To calculate the pressure distributions and wave resistances at different speeds of the Wigley hull, the Series 60 hull and the S175 container ship hull, then compare the results with available theoretical and experimental data available in the literature.
Organization of the Thesis
To study the uncertainty analysis of solution, a systematic variation of the number of control volumes of structured grid will be performed for hydrofoils. iv). The solution methodology incorporates incomplete LU decomposition, momentum interpolation and the SIMPLE Algorithm on structured curvilinear composite grids.
An efficient solution procedure, which facilitates the use of the non-staggered variable arrangement, has been developed. The turbulence model k-ω SST presented the best prediction of the flow properties for the obstacle among the investigated turbulence models.
Conventional Time Averaging
3.1.6), where t is not large compared to the period of random fluctuations associated with turbulence, but small relative to the time constant for any slow changes in the flow field associated with ordinary non-uniform flow. In practice, ∞ means a time that is long compared to the reciprocal of the dominant frequencies in the spectrum 𝑓.
Reynolds Averaged Navier-Stokes Equations
- Reynolds form of continuity equation in time averaged variables
- Reynolds form of momentum equation in time averaged variables
The conservation form of momentum equations in x, y and z directions for a viscous flow is given by The time-averaged momentum equations in the y and z directions can be obtained using a similar methodology.
Closure Problem for Incompressible Flows
Introduction
Turbulence Models
Can be used for transition flows (although it tends to predict an early transition). Disadvantages of the k-ω model are: SST (Menter's Shear Stress Transport). Shear Stress Transport (SST) is a variant of the standard k–ω model.
K-ε Two Equation Turbulence Model
This introduces six new unknowns (components of the tensor 𝜌𝒗̅̅̅̅̅̅′𝒗′ known as the Reynolds stress tensor 𝜏𝑅) in the momentum equations. And ε denotes the dissipation of turbulent kinetic energy given by 𝜀 = 𝑣 {(𝜕𝑢 The physical meaning of the above equations is: the rate of change and advective transport k or ε is equal to the diffusive transport together with the rate of production and destruction k or ε.
Governing Equations
Boundary Conditions
From the above parameters it is clear that the input values of u-velocity, v-velocity, w-velocity, turbulent kinetic energy and dissipation must be provided. At the surface of the body: For stationary walls, the default consideration is to assume that the slip condition applies, which simply means that the velocities are assumed to be zero at the rigid boundaries.
Generic form of Governing Equations
Grid Classification
- Structured curvilinear grids
- Unstructured grids
Compared to a structured network, the storage requirements for an unstructured network can be significantly greater, since the neighbor connection must be stored explicitly. The advantage of this type of grid is that the mesh can be refined where necessary.
Staggered vs Non-staggered Grid
The numerical formulation presented in this paper is based on a lumped arrangement of the Cartesian velocities where the velocity components and the pressure are located at cell centers. Grouped arrangement of the primitive variables has some clear advantages over staggered grids, especially when non-orthogonal coordinates are used for the simulation of complex geometries.
Transformation of Governing Equation
The detailed calculation of the transformation of convection and diffusion terms using the transformation rules is described in Appendix C.
Finite Volume Discretization in Structured Curvilinear Grid
We can achieve this by means of a weighting parameter θ between 0 and 1 and write the integral of 𝛷 as. Where asterisk is used on quantities calculated from previous iteration. Similarly for turbulent dissipation, the source term is written as The terms multiplied by ψ become part of the source term and are explicitly determined from continuous iterations.
The same is done with cross-diffusion terms: these are also evaluated on each CV page, summed and added to the source term.
- Introduction
- Incomplete LU Decomposition: SIP Solver
- Momentum Interpolation
- The SIMPLE Algorithm on Structured Curvilinear Collocated Grids
𝚽} is a column matrix having values K 𝛷 and {S} is the matrix of the same form containing terms K S. 63 Fig.8.2: Graphic representation of the coefficient matrix for the three-dimensional problem In Fig.8.2 the structure of the coefficient matrix [A] is presented for the case when, K = Ni x Nj xNk , where Ni , Nj and Nk are the number of nodes in each coordinate direction. The order of the nodes is such that the index j (see Fig.8.1) first increases, then i and then k.
The second approach is believed to be more appropriate, as slow changes in field values can sometimes be misleading (eg changes can be small due to low values of .. 71 under-relaxation factors used).
Implementation of Boundary Conditions
9.1, the inlet boundary is the west side of the cell with center P. The top and bottom of the control volume are in the perpendicular direction of the two-dimensional cell. If the values of the dependent variables are known at the output limits, they are recorded as such. In the case of the speed components, it is also necessary to ensure that they comply with overall continuity, i.e.
It is known that the values of the velocity components (u, v, w) and therefore the mass fluxes 𝐹𝑒, 𝐹𝑤, 𝐹𝑛, 𝐹𝑠, 𝐹𝑡, 𝐹𝑏 are zero there.
Procedure of Resistance Calculation
- Calculation of unit normal vector in free streamline direction of a cell
- Calculation of cell surface area
The area of the cell, dS, can be found by the absolute value of the cross product of a⃗ and b⃗ multiplied by 12, as shown in the figure. Using the pressure integration method over the entire surface of the underwater body, the resistance is calculated for the Wigley hull, the Series 60 shape hull and the S 175 container ship hull.
NACA 0012 Hydrofoil
However, the results obtained show inconsistency at the leading edge at 10-degree angle of attack compared to the numerical result of Mursaline (2017). For the 198x40x28 grid, effect of angle of attack is studied by plotting the pressure coefficients at 0, 6, 10 and 15 degrees on the same axes as shown in Fig. 11.9. the experimental result.
The grid uncertainty results for the lowest pressure coefficient of NACA 0012 at x/c=0.05 for 15 degree angle of attack.
NACA 4412 Hydrofoil
The obtained results show some discrepancy at the leading edge at a lower angle of incidence compared to the numerical result of Mursaline (2117), but at a higher angle of incidence the obtained result is in very good agreement with the numerical result of Mursaline (2017). 11.23, for the NACA 4412 (circular hydrofoil), at a zero degree angle of attack, the static pressure is slightly higher on the lower surface than on the upper surface, creating lift. The obtained results for the lift coefficients for different angles of incidence are in good agreement with the numerical result of Mursaline (2017).
Pressure coefficients at a given angle of attack are plotted for three grid sizes on the same axes at Re = 3.1 x 106.
Wigley Hull
It is found that the positive pressure values are concentrated on the small areas in the vicinity of the bow and the stern of the hull, and the negative pressure values occur on the rest of the hull. At the bow and stern of the hull, fluid velocity is lower, so pressure is higher according to Bernoulli's theorem. On the other hand, the fluid velocity is higher around the middle section of the hull, so the pressure is lower.
Finally, using the pressure distribution along the length of the hull at different Froude numbers, drag coefficients (Cw) for Wigley hull are calculated and compared with the numerical result obtained by Tarafder and Suzuki (2007) and experimental results reported by Shearer and Cross was performed. (1965).
Series 60 Hull
It is believed that the deviations from the numerical and experimental results at high Froude numbers occur due to the shift of peak and valley values in the pressure distribution and also due to insufficient interference of the hull surface during the drag calculation. 116 Comparisons of the pressure distribution at the waterline, bilge and keel along the length of the hull are shown for the Froude number, Fn=0.20 in figure. Graphs have also been created for the pressure distribution along the length of the Series 60 hull at the waterline, bilge and keel. keel section in the same graph.
It is believed that the deviations from the numerical and experimental results at high Froude numbers occur due to the shift of peak and valley values in the pressure distribution and also due to insufficient interference of the hull surface during the drag calculation.
S175 container ship hull
Finally, using the pressure distribution along the length of the hull at different Froude numbers, drag coefficients (Cw) are calculated for the series 60 hull and compared with the numerical result obtained by Tarafder and Suzuki (2007) and the experimental result operated by Ishikawajima-Harima Heavy Industries Co., Ltd. At Froude number 0.20 the pressure distributions over the length of the S175 container ship hull at the waterline, the bilge and keel section are shown in Figure 120. At Froude number 0.28 the pressure distributions over the length of the S175 are shown. The hull of the container ship at the waterline, the bilge and keel section are shown in Fig.
Finally, using the pressure distribution along the length of the hull at different Froude numbers, resistance coefficients (Cw) are calculated for S175 container ship hull.
Conclusion
An increase in the coefficient of resistance to creating waves (Cw) is usually accompanied by a number of bumps and hollows in the resistance curve of the ship's hull. The obtained Wigley hull wave resistance coefficients show small discrepancies at relatively low Froude numbers, i.e., Fn, but the discrepancies are large at relatively high Froude numbers, i.e., Fn. The calculated resistance coefficients for the Series 60 hull form waves show small deviations at relatively low Froude numbers, i.e., Fn, but the deviations are large at relatively high Froude numbers, i.e., Fn .
The resulting wave-making resistance curve of the S175 container ship hull follows the trend of the Series 60 hull shape resistance curve.
Recommendation
2] Benahmed, L., and Aliane, K., "Simulation and Analysis of a Turbulent Flow Around a Three-Dimensional Obstacle," Acta Mechanica et Automatica, Journal of the Technical University of Bialystok, vol. 4] Bouhelal, A., Smaili, A., Guerri, O., and Masson, C., "Numerical investigation of the turbulent flow around a horizontal end axis wind turbine using low and high Reynolds models" , Journal of Applied Fluid Mechanics, Vol. 32] Orihara, H ., and Miyata, H., “Estimation of added resistance in ordinary incident waves by computational fluid dynamics motion simulation using an overlay.
33] Orihara, H., and Miyata, H., “A Numerical Simulation Method for Flows Around a Transverse-Stern Semiplaning Boat”. 40] Rhie, C.M., and Chow, W.L., “A Numerical Study of Turbulent Flow Past an Airfoil with Trailing Edge Separation”. 45] Saha, G., and Tarafder, M.S., "Calculation of Flows Around a Stern Hull by a Modified Rankine Source Panel Method", Journal of Mechanical Engineering, Vol.
