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A new aerodynamic force of drag prediction approach was developed to compute the airfoil drag via entropy generation rate in the flow field. According to the momentum balance, entropy generation and its relationship to drag were derived for viscous flow. Model equations for the calculation of the local entropy generation in turbulent flows were presented by extending the RANS procedure to the entropy balance equation. The accuracy of algorithm and programs was assessed by simulating the pressure coefficient distribution and dragging coefficient of different airfoils under different Reynolds number at different attack angle. Numerical data shows that the total entropy generation rate in the flow field and the drag coefficient of the airfoil can be related by linear equation, which indicates that the total drag could be resolved into entropy generation based on its physical mechanism of energy loss.

Accurate prediction of the aerodynamic force is a critical requirement in aircraft design. CFD methods have been widely applied in aerodynamic design and optimization of aircraft. However, even for airfoils with attached flow at relatively low angles of attack, the predicted drag based on integration of the surface pressure and skin-friction distributions can be off by more than 100% even though the computed surface pressure and skin friction are in good agreement with the experimental data [

There are two common approaches to predict total drag force of airfoils or wings, a standard surface integration method, and a wake integration method. The surface integration method relies on calculations of pressures and skin friction over a series of flat surfaces. However, the surface integration was met with difficulties especially for the complex configuration due to the need to approximate the curved surfaces of the body with flat faces which can be affected by significant errors introduced by the “numerical viscosity” and “discretization” error of the numerical solution^{} [

Oswatitsch [

It is pointed out that a representation of losses in terms of entropy generation offers significant insight into the flow and thermal transport phenomena over the airfoil and provides an effective tool for drag prediction. Entropy analysis is a method to evaluate a process based on the second law of thermodynamics. It is basically calculating entropy generation in a system and its surroundings and using it as a proxy for the evaluation of the energy loss [

In this paper entropy generation and its relationship to drag were derived for viscous flow. we present model equations for the calculation of the local entropy generation rates in turbulent flows by extending the RANS procedure to the entropy balance equation. This equation serves to identify the entropy generation sources, without need to solve the equation itself. The main objective of this paper is to demonstrate the viability of entropy-based drag calculation method and to compare the consistency of predicting the drag of single-element airfoils using surface integration, wake integration, and entropy generation integration.

The conservation law of momentum to the control volume enclosing the configuration makes it possible to predict the total aerodynamic drag force by an integration of stresses on the aircraft configuration (surface integration) or by an integration of momentum flux on a closed surface far from the configuration (wake integration). We consider a steady flow with freestream velocity

The far-field expression of drag prediction is given by the right-hand side of (

The wake expression for drag is derived from (

In a two-dimensional flow, as we move the boundary

For ideal gas, the module of the velocity can be expressed in terms of variations of total enthalpy (

Equation (

After substitution of expression (

The term depending on

Finally, (

According to Gouy-Stodola theorem, the relationship between the exergy destruction and the entropy generation is defined by

The entropy is a state variable and the transport equation for entropy per unit volume in Cartesian coordinates can be expressed as

There are basically two methods how entropy generation can be determined [

In the indirect method, the entropy generation is calculated by equating it to the rest of (

Obviously the direct method is superior and should be applied in complex flow situations. And, there is one more important advantage of this method [

When the turbulent flow is considered, the derivation of the entropy generation rate is carried out in terms of the RANS equations which splits the velocities ^{}[

In the flow field calculation, an in-house code was used. The RANS based turbulence models are used in conjunction with the Navier-Stokes equations for viscous flow simulations. The numerical simulations discussed herein use the general steady viscous transport equations in conservative form which can be casted into the following compact notation form:

We discrete the transport equations by FVM (Finite Volume Method) on a nonorthogonal collocated grid that all transport variables are stored at cell centers and the integration and discretization about the control volume

The deferred correction method [

The diffusion term at the face is

For evaluating

For steady flows, strongly implicit procedure (SIP) [

So far, we have shown how the total entropy generation in turbulent flows can be calculated in a postprocess. In this section, entropy generation calculation has been carried out using a turbulent flow over NACA standard series airfoils to validate the eddy viscosity model used for entropy generation calculations in the present paper, details of which are provided below. We discussed the relationship between the total entropy generation in the flow field and the drag coefficient at various angle-of-attack under different Reynolds number with different turbulence models.

The nonorthogonal structured mesh is generated by algebraic method. As shown in Figure

Computational domain around the airfoil: (a) complete grid and (b) fine grid around airfoil.

Shuja et al. [

The surface pressure distribution at different angle-of-attack compared with the experimental data^{} [

Surface pressure coefficient distribution profiles for NACA0012 airfoil at different turbulence models.

Contours of entropy generation rate around the NACA0012 airfoil for angle-of-attack

Streamlines and contours of

Figure

Entropy generation rate (a) and drag coefficient (b) versus angle-of-attack for NACA0012 airfoil at different turbulence models.

Figure

Drag coefficient versus entropy generation rate at different turbulence models.

To study the relationship between the Reynolds number effect and the entropy generation, we simulated the flow over NACA0012 airfoil under different Reynolds number with

Streamlines and contours of

Figure

Entropy generation rate distribution along airfoil for attack angle 0° (a), 10° (b), and 20° (c) at different Reynolds number.

Figure

(a) Entropy generation rate and (b) drag coefficient versus angle-of-attack for NACA0012 airfoil at different Reynolds number.

We consider a dimensionless version of (

Equation (

Drag coefficient versus entropy generation rate for NACA0012 at different Reynolds number.

Moreover, in order to study the relationship between the airfoil configuration and the entropy generation, we simulated the flow over different standard series NACA airfoils under

(a) Entropy generation rate and (b) drag coefficient versus angle-of-attack for different airfoils.

Figure

Drag coefficient versus entropy generation rate for different airfoils.

In this section, the entropy generation and its relationship to drag is validated by numerical simulating the total entropy generation rate and drag coefficient of different airfoils under different Reynolds number at different attack angle. Therefore, we are confident to say that entropy generation in turbulent flow using the effective viscosity leads to a good approximation of aerodynamic drag of airfoil.

Next we chose NACA0012 airfoil as the benchmark airfoil as it is well documented in the open literature. By comparing drag from alternative numerical methods, we are able to verify and validate the computational procedure of airfoil drag based on entropy generation method. The free stream conditions are set to be

Computational domain around the NACA0012 airfoil: (a) complete grid, (b) fine grid, and (c) deformed grid around airfoil.

In order to ensure that the numerical model is free from numerical diffusion and artificial viscosity errors, several grids are tested to estimate the number of grid elements required to establish a grid independent solution. Table

Specification of grid and convergence study (

Grid | Number of elements | First cell wall distance | Growth rate | | | | | |
---|---|---|---|---|---|---|---|---|

Coarse grid | | | 1.15 | 50~200 | 124 | 39 | 85 | 0.804 |

Medium grid | | | 1.15 | 5~40 | 103 | 23 | 80 | 0.919 |

Medium-fine gird | | | 1.15 | <5 | 94 | 21 | 73 | 1.352 |

Fine gird | | | 1.05 | <1 | 101 | 24 | 77 | 1.385 |

The nondimensional normal distance from the first grid point to the wall along the airfoil.

Pressure coefficient plotted on the normalized airfoil cord for different grid resolutions.

Using the obtained velocity field, drag force on the airfoil from entropy generation is calculated and compared with surface and wake integration methods. The drag can be expressed in the following forms:

Table

Comparison of drag coefficients from various methods (NACA0012 airfoil,

Angle-of-attack | | Drag from experiment | Drag from surface integration | Drag from wake integration | Drag from entropy generation | |||
---|---|---|---|---|---|---|---|---|

| | | | |||||

| 1.35 | 80.0 | 94.2 | 95.3 | 93.4 | 92.4 | 92.1 | 89.1 |

| 1.53 | 90.0 | 112.7 | 110.5 | 109.2 | 107.6 | 107.1 | 101.0 |

| 2.23 | 120.0 | 177.8 | 166.3 | 164.5 | 163.0 | 162.4 | 147.2 |

| 4.15 | 190.0 | 346.3 | 312.2 | 311.5 | 310.3 | 309.6 | 273.9 |

Figure

NACA 0012 airfoil. Computed drag coefficient versus angle-of-attack.

Traditionally, surface integration of the pressure and stress tensor on the body surface of aircraft, which is called surface integration, is used for the drag prediction in CFD computations. But it is pointed out that the drag computed by the near-field method has inaccuracy relating to the numerical diffusion and error. An advanced drag prediction method (wake integration) based on the momentum conservation theorem around aircraft is watched with keen interests which can compute the drag components from the surface integration on the wake plane of the downstream of aircraft. The method has the drag decomposition capability into wave, profile, and induced drag component. However, the method is closely related to the wake cross surface. In this paper, a drag prediction method based on entropy generation was integrated into a RANS-based CFD solver and an approach was developed to compute the airfoil drag via entropy generation rate in the flow field which is a volume integral method derived from far-field method by applying the divergence theorem of Gauss. Present paper has been devoted to the analysis of some RANS calculations around two-dimensional airfoils, at subsonic freestream conditions. The main objective of this paper is to compare the consistency of predicting the drag of single-element airfoils using surface integration, wake integration, and entropy generation integration. Overall, entropy generation integration shows potential as a simpler method than surface integration and wake integration for calculating drag. The main advantage of this technique is that no detailed information on the surface geometry of the configuration is needed. The results show that drag prediction using CFD and entropy generation integration is possible within engineering accuracy and that the proposed method also have the drag visualization capability in the flow field for designers to take measures to minimize drag. The main conclusions are summarized as follows:

The drag and entropy generation in 2D domain can be related by a linear balance equation, so the drag of the airfoil can be directly estimated from total entropy generation without other effects.

Entropy generation consists of two parts: heat transfer and viscous dissipation. In turbulent flow, the fluctuating velocity and temperature contribution can be accounted for by using the effective viscosity and effective thermal conductivity.

In the 2D turbulent flow of airfoil, most of the entropy is generated in the front region, the near wall region, and the turbulent wake region. The total entropy generation of flow field increases with the angle-of-attack and Reynolds number.

Drag prediction using CFD and entropy generation integration is possible within engineering accuracy. Future work should concentrate on 3D airfoil drag and compressible turbulent flow to demonstrate the universality of entropy-based method in CFD prediction of airfoil drag.

The authors declare that they have no conflicts of interest.