Study of the physical interpretation of the volume term in the unsteady induced drag component

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To account for all the additional terms due to unsteadiness and relate them to phenomenological components

It may seem obvious but in order to be certain not to forget any additional term due to the unsteadiness of the flow, one should start back from the balance of mass and momentum equations in order to obtain the correct far-field equation. Trying to add time derivative terms in the expression of the total far-field drag would lead to a wrong starting point.
Furthermore, these additional terms, once correctly accounted for, must be attributed to existing drag components, or to new ones. There are a few studies in the literature concerning unsteady drag in incompressible multiphase cases, which can be applied to moving airships for example. A so-called virtual mass force, first identified by Prandtl [69], and a Basset force [4] have been defined as resulting from the acceleration of the body.
However, nothing was found in the literature for transonic cases which we are interested in. The first and only team who has tackled the question of unsteady drag breakdown [31] has chosen to create a so-called unsteady drag coefficient, which is not satisfactory on the physical and phenomenological point of view.
The difficulty will therefore be to identify the physical source of this additional drag. Is it an unsteady contribution into each drag component, such as the contribution of a moving shock in a buffet case? Are there new phenomena which cause drag when the flow is unsteady, such as acoustics? Can it be considered a phenomenological component of drag?

To avoid applying the steady theory as such

We have now settled that the demonstration for the steady theory should be adapted to unsteady flows from the very beginning, that is to say the derivation of the far-field equation. Now another obstacle may arise in the following steps of the demonstration. The assumptions used in the steady
theory, which can have been used without being clearly mentioned in the original paper from Van der Vooren and Destarac [90], may indeed be inapplicable to unsteady flows. All the assumptions made will therefore have to be pinpointed and adapted to unsteady flows.
Another point is that we need to ponder in the unsteady framework, when we are rather used to thinking in a steady framework. It may for example seem unnatural that a snapshot at a given time of the flow around an object can be sufficient to give the force applied by the fluid on the body at the exact same instant. This is however what the momentum and mass balance equations tell us. The
information which allows to account for propagation delays and history of the flow is provided by the time derivative terms as we will see later.

To take the propagation delays into account

The last obstacle consists in taking correctly into account the propagation in time and space, which imply delays in the transmission of perturbations. The history of the flow must also somehow be taken into account to correctly assess the force experienced at a given instant by the body. Remember that the principle of far-field methods consists of integrating phenomena occurring in the flow over a control volume.
A concrete example will make it more clear. Let us consider an airfoil in a flow field, with a shock wave on the upper surface. The viscous effects are assumed to be one order higher than those due to the shock wave and its motion, and are therefore neglected. A streamtube encloses the wave, as depicted in Figure I.1(a) at time t = ti. Now let us consider a perturbation of this shock wave, say that it suddenly moves downstream at time t = ti + t as sketched in Figure I.1(b). This perturbation of the shock wave induces a perturbation of the flow which is shed downstream in the wake of the wave, inside the streamtube, such as in Figure I.1(c).

Derivation of a directly generalizable proof of Van der Vooren’s steady formulation

Van der Vooren’s formulation allows to split the drag into three phenomenological components: wave, viscous and induced drag, using a thermodynamic breakdown. A spurious contribution is also assessed. It accounts for a part of the numerical drag due to numerical dissipation and can be of great interest to characterize the quality of a mesh or to optimize a configuration with a poor mesh quality. This steady formulation has proven its robustness and precision in many industrial cases and has been developed conjointly by Van der Vooren and Destarac at ONERA for about 20 years. A historical review of the existing drag prediction methods is given in the Chapter Presentation of the main existing methods of numerical drag prediction.
The idea in this section is to give a new derivation of Van der Vooren’s theory [90], since some steps are missing in the original demonstration and restrictive assumptions regarding the extension to unsteady flows are used. There are three main steps in the method: the derivation of the farfield equation from conservation equations, the thermodynamic breakdown into induced and profile components, and the volume splitting of the profile component into a wave and a viscous component.
The approach adopted here keeps the same process, but differs as far as the volume splitting is concerned. All steps will however be demonstrated in details in order to ensure that no hypothesis is forgotten. Practical refinements will then be presented.

Criteria used for the definition of the integration surfaces

The theory as derived here requires the definition of streamtubes as integration surfaces for the wave and viscous drag components. The induced surface is in the raw version defined as the outer surface of the control volume. In practice, it has been observed that smaller surfaces imply better robustness. It also has consequences on the spurious drag coefficient, which has been observed to be smaller with smaller surfaces. Moreover, streamtubes are not easily constructed in a postprocessing tool. Physical criteria are consequently used in practice, leading to small discrepancies with the theoretical definitions. The corresponding surfaces will be denoted with a superscript p for practice.
We have seen with our definition that the downstream face of the streamtube enclosing the shock could be moved close to the shock wave. Since the surface is quite small around the shock wave, and since the contributions over lateral surfaces will remain small whatever their shape, a physical criterion of shock detection is in practice used to define the integration surface.
This criterion was first used by Tognaccini [83]. It allows to detect the presence of a shock wave from the comparison between the local velocity in the direction of the pressure gradient and the speed of sound: if this velocity is greater than the speed of sound, then the cell must be inside the shock integration surface. In practice, a coefficient cshock close to 1 is added. It is chosen equal to 0.95 in most cases but can be tuned by the user. q ·∇p ≥ cshock a k∇pk ⇒ cell ∈ Spw.

Generalization of Van der Vooren’s formulation to unsteady flows

The extension to unsteady flows of Van der Vooren’s theory is not as straightforward as it seems. Where some authors [31] chose to define an unsteady drag coefficient concentrating all unsteady contributions, we have finally decided to allocate unsteady terms to each drag component, in order to take the early drag creation and propagation in the fluid into account. The main difficulty tackled here is indeed the synchronization of the fluid phenomena and the loading experienced by the body at the same instant.
In order to do so, we have derived a new proof of Van der Vooren’s formulation and carefully located all the assumptions which have been made:
• The unsteady terms in the balance of mass and momentum equations were removed.
• During the thermodynamic breakdown step, under the hypothesis of an only irreversible flow, it
was assumed that the pressure went back to the reference pressure and the velocity was parallel to the reference velocity in a downstream wake plane.
• At the volume splitting step, it was assumed that the flow was not affected by the presence of the shock wave or the boundary layer outside the streamtube enclosing the source.
• When moving the integration surfaces, it was assumed that the flow was isentropic and isenthalpic downstream of a shock wave.

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Table of contents :

Nomenclature
General Introduction
Presentation of the Main Existing Methods of Numerical Drag Prediction
1 Historical presentation of the main thermodynamic methods
1.1 Steady methods
1.1.1 Betz
1.1.2 Jones
1.1.3 Oswatitsch
1.1.4 Maskell
1.1.5 Van der Vooren and Destarac
1.2 First unsteady generalization of Van der Vooren’s formulation
1.2.1 Theoretical developments
1.2.2 Results on a pitching case
2 Presentation of the formulations based on the velocity vector
2.1 Formulation for steady incompressible flows
2.2 Breakdown into induced and profile components for steady incompressible cases .
2.3 Extension to steady compressible flows
2.4 Breakdown in the steady compressible case
2.5 Generalization to unsteady flows
2.5.1 Noca
2.5.2 Wu
2.5.3 Marongiu
2.5.4 Xu
2.5.5 Other contributions to the unsteady generalization
I Development of an Unsteady Formulation starting from Van der Vooren’s Formulation
1 What is difficult about a generalization to unsteady flows?
1.1 To account for all the additional terms due to unsteadiness and relate them to phenomenological components
1.2 To avoid applying the steady theory as such
1.3 To take the propagation delays into account
2 Derivation of a directly generalizable proof of Van der Vooren’s steady formulation
2.1 Derivation of the far-field equation in the steady case
2.2 Thermodynamic breakdown
2.2.1 Breakdown of vector f
2.2.2 Derivation of the irreversible axial velocity
2.3 Volume splitting using streamtubes
2.3.1 Wave drag
2.3.2 Viscous drag
2.3.3 Another justification of the use of streamtubes
2.4 Derivation of the final steady formulation
2.4.1 A first « raw » formulation
2.4.2 Numerical deviations from the theory
2.4.3 Practical refinements of the theoretical formulation
3 Generalization of Van der Vooren’s formulation to unsteady flows
3.1 Implementation of the additional unsteady terms in the far-field equation
3.2 Derivation of the four components unsteady formulation
3.2.1 Unsteady wave drag expression
3.2.2 Unsteady viscous drag expression
3.2.3 Unsteady induced drag expression
3.2.4 Final decomposition
3.3 Criteria used in practice for the integration volumes definition
4 Discussion
4.1 Robustness of the formulation
4.1.1 Domain of definition of the irreversible axial velocity
4.1.2 Physical criteria used for the definition of the integration volumes
4.2 Physical background for the definition of the unsteady induced drag
4.3 Comparison with Gariépy’s formulation
II Study of Improvement Axes for the Robustness and the Physical Background 
1 Study of an alternative expression for the irreversible axial velocity to improve the robustness
1.1 Derivation of the expression developed by Méheut
1.2 Domain of definition of the reversible axial velocity
1.3 Study of its theoretical validity
1.4 Analysis of the variant suggested by Gariépy
1.5 Comparison of the three expressions on several steady test cases
1.5.1 Airfoil in a transonic inviscid flow: assessment of CDw
1.5.2 Airfoil in a subsonic viscous flow: assessment of CDv
1.5.3 Wing in a subsonic inviscid flow: assessment of CDi
1.5.4 Wing in a transonic viscous flow: assessment of all three drag components
2 Study of new criteria for the robustness of the volume definitions
2.1 Expression of the unsteady criterion
2.2 Evaluation on an unsteady subsonic test case
2.3 Evaluation on an unsteady transonic test case
2.4 Filtering
2.5 Conclusions on the validity of the unsteady wave criterion
3 Study of the physical interpretation of the volume term in the unsteady induced drag component
3.1 Link between surface and volume terms
3.2 Acoustic effects
3.3 Breakdown of the unsteady induced drag component
4 Description of the final method used for the unsteady applications
4.1 Final formulation with five components
4.2 Good practice recommendations
III Assessment of the Wave, Viscous, and Acoustic Drag Components on Naturally Unsteady Cases 
1 Application to a vortex shedding case
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
1.7 Comparison with Gariépy’s formulation
1.8 Comparison between steady and time-averaged unsteady results
2 Application to a buffet case simulated by a URANS method
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Comparison with Gariépy’s formulation
2.8 Comparison between steady and time-averaged unsteady results
3 Conclusions regarding the validity of the method
IV Assessment of the Motion, Induced, and Propagation Drag Components on Mobile Cases 
1 Application to a pitching airfoil in an inviscid flow
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
1.7 Comparison with Gariépy’s formulation
1.8 Influence of the reduced frequency
1.9 Comparison between steady and time-averaged unsteady results
2 Application to a pitching airfoil in a viscous flow
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Comparison with Gariépy’s formulation
2.8 Influence of the reduced frequency
2.9 Comparison between steady and time-averaged unsteady results
3 Conclusions regarding the validity of the method
V Application of the Unsteady Formulation to Complex Cases 
1 Application to a pitching wing in an inviscid flow
1.1 Quick literature review
1.2 Description of the test case
1.3 Convergence study
1.4 Analysis of the flow field resulting from the simulation
1.5 Application of the drag extraction method
1.6 Analysis of the drag breakdown results
2 Application to a buffet case simulated by the ZDES method
2.1 Quick literature review
2.2 Description of the test case
2.3 Convergence study
2.4 Analysis of the flow field resulting from the simulation
2.5 Application of the drag extraction method
2.6 Analysis of the drag breakdown results
2.7 Spectral analysis
2.8 Comparison with URANS results
General Discussion
Conclusion and Perspectives
Appendices
A Numerical tools used for the applications
A.1 Modeling of aerodynamics
A.1.1 RANS approach
A.1.2 LES Approach
A.1.3 Hybrid RANS/LES approaches
A.2 Codes used
B Grid studies
B.1 Airfoil in a steady transonic inviscid flow
B.2 Airfoil in a steady subsonic viscous flow
B.3 Wing in a steady subsonic inviscid flow
B.4 Pitching airfoil in a viscous flow
B.5 Pitching wing in an inviscid flow
C Time evolution figures
Bibliography 

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