WP3

Main Achievement

WP3 – Flow stability and adjoint

Objectives: To develop numerical tools for stability and receptivity analysis of two- (2D) and three-dimensional (3D) laminar and turbulent 3D complex flows

Description of Work and Role of Partners

In this WP, we make a distinction between those partners more interested in fundamental boundary layer transition problems (T3.1) and those more focused on phenomena related to compressible complex detached flows (T3.2).

A more detailed description of the activities is included in deliverables D3.1 to D3.4

T3.1 Stability of boundary layers under surfaces irregularities.

The stability of boundary layers under surface irregularities is crucial in fluid dynamics, especially in determining flow behavior and drag forces. When a fluid flows over a surface with irregularities (such as roughness or undulations), these disturbances can affect the boundary layer’s development and stability. Irregularities can lead to earlier transition from laminar to turbulent flow, increasing friction and resistance. The interaction between surface roughness and the boundary layer can result in separation or reattachment, influencing flow patterns and performance. The stability depends on the size, shape, and distribution of the surface irregularities, as well as the Reynolds number of the flow.

In this regard:

ESR9 developed a linear stability tool capable of analysing the evolution of secondary instabilities in crossflow-dominated boundary layers and the effect of surface irregularities in swept-wing flows. The methodology relies on the linearized stability equations in generalized curvilinear non-orthogonal coordinate systems. Different types of surface irregularities were analyzed, such as forward-facing steps, smooth surface humps, and roughness elements. Results regarding the impact of cylindrical roughness elements were presented in D3.3 (‘The effect of surface irregularities on laminar-turbulent transition’). Two-dimensional Linear Stability Theory (LST-2D) and three-dimensional Parabolized Stability Equations (PSE-3D) were used to study the impact of these surface imperfections. Comparisons with Direct Numerical Simulation (DNS) results showed that the linear stability methodologies correctly reproduce the shape and location of the perturbation. Moreover, it was noted that the PSE-3D approach displays good agreement with DNS results in terms of the streamwise evolution of the integrated growth rates, whereas the LST-2D approach underestimates the growth rates. An example is shown in the following figures, which compare the methodologies for the most amplified mode downstream of a smooth surface hump geometry

Figure 1: Illustrative comparison of the methodologies for the most amplified mode downstream of a smooth surface hump geometry

Additionally, ESRs 9, 10, and 15 collaborated on the numerical investigation of the effect of surface irregularities on the transition to turbulence using a DNS code. ESR 10 coordinated the effort to produce deliverable D1.5. ESR 10’s individual contribution to the deliverable included the receptivity analysis of an incompressible, two-dimensional boundary layer subjected to free-stream turbulence in the presence of a discrete roughness element. DNS simulations were performed for several flow configurations over a flat plate with a semi-elliptic leading edge, comparing the evolution of perturbations in each case.

The effect of discrete roughness elements (DRE) mounted on a flat plate with a semi-elliptical leading edge was investigated by studying the evolution of the perturbation amplitudes along the flat plate surface. The height of the DRE was used as a control parameter. The receptivity analysis examined the interaction between the DRE and two different types of perturbations: first, a forced, monochromatic Tollmien-Schlichting wave generated by a suction-blowing slot over the flat plate surface, considering two relevant wave frequencies; and second, isotropic free-stream turbulence imposed at the domain’s inlet, where two cases with different turbulence intensity values were analyzed.

In the same line, ESR15 is focused on implementing the effects of surface irregularities in the linear and parabolized stability tools and applying these tools to laminar-turbulent transition cases, contributing to D3.4. ESR15 has developed a physics-based numerical framework to determine manufacturing tolerances such that early transition, compared to clean airfoils, does not occur for NLF airfoils in the presence of smooth surface waviness. Two optimization algorithms have been employed, and their results are compared across a wide range of Mach numbers and Reynolds numbers.

T3.2 Stability of general detached flows.

The stability of general detached flows, such as those occurring in separated boundary layers or wakes, is a key topic in fluid dynamics. Detached flows involve regions where the fluid no longer follows the surface, leading to flow separation. This can result in unsteady, turbulent flows with complex behavior. Stability analysis in such flows focuses on understanding the transition from laminar to turbulent states, driven by instabilities such as shear-layer, Kelvin-Helmholtz, or global modes. The interaction between these instabilities and the flow’s geometry, pressure gradients, and boundary conditions can significantly impact drag, heat transfer, and overall aerodynamic performance.

ESR2 investigates the laminar boundary layer separation over a wall-mounted bump geometry at low Reynolds numbers to understand the dynamics of transition in separated flows under pulsating inflow conditions. This study, part of Test Case 2 (TC2), utilizes the bump geometries from Purdue University’s experimental study. The primary objectives are to explore how pulsating inflow influences the separation process, transition to turbulence, and the length of the separated region. Direct Numerical Simulations (DNS) are conducted to analyze the effects of harmonic inflow on the boundary layer separation and transition processes, with a focus on the acceleration parameter (K) and incoherent spanwise vorticity.

The results show that under weak pulsating inflow, the separation process is dominated by the continuous shedding of Tollmien-Schlichting (KH) vortices, followed by a rapid transition to turbulence and recirculation. The separated shear layer gradually moves in response to changes in the bulk velocity, and the dynamics resemble steady inflow cases. In contrast, strong pulsating inflow leads to the formation and release of a large vortex cluster, which is synchronized with the harmonic changes in bulk velocity. This scenario shows a marked departure from KH vortex shedding. The medium pulsating inflow scenario exhibits a combination of weak and strong pulsating inflow dynamics, where KH vortices and vortex clusters intermittently appear.

The study also includes sensitivity analysis based on the acceleration parameter (K), with a focus on the minimum phase-averaged component along the bump surface. This analysis helps quantify the impact of pulsating inflows on reducing the separated region and creates a sensitivity map that links harmonic inflow intensity to separation behavior. The findings provide valuable insights into vortex dynamics and transition processes in separated boundary layers under pulsating inflow conditions, with implications for aerodynamic applications involving flow separation.

ESR8 developed, validated, and applied a general stability analysis toolbox based on a high-order discontinuous Galerkin framework. To achieve this, ESR8 created numerical tools from scratch using FreeFem++ to study the stability and receptivity of both 2D and 3D turbulent flows. These models are based on linearizing the Reynolds-Averaged Navier-Stokes (RANS) equations. The methodology was successfully applied to investigate the stability of complex detached (or separated) flows, such as flow around a stalling airfoil or a turbulent separation bubble. The study identified the mechanisms behind complex aerodynamic 2D and 3D phenomena, including the formation of stall cells and low-frequency oscillations.

This toolbox was used to study free-transitional transonic buffet around the OALT25 profile, where low-frequency shock oscillations were linked to the presence of an unstable eigenvalue in the mean-flow Jacobian, consistent with classical fully turbulent scenarios. Additionally, resolvent analysis revealed three distinct instability mechanisms: laminar separation bubble instability, vortex shedding, and Kelvin-Helmholtz-type instabilities. The results of this in-depth analysis showed good agreement with previous experiments and high-resolution LES data.

Figure 6: Left: Direct density eigenvector associated with the unstable transonic buffet eigenvalue around the OALT25 profile. Right: Resolvent response mode to the optimal forcing for a vortex-shedding like phenomenon around the OALT25 profile.

This stability approach was further adapted for fluid-structure interaction by incorporating a structural model with up to two degrees of freedom and coupling it with a moving reference frame formulation of the RANS equations. The method was successfully integrated into the existing framework and extensively validated. This integration enabled a detailed aeroelastic analysis of the OALT25 airfoil near transonic buffet, leading to the identification of multiple distinct aeroelastic patterns through both linear and nonlinear unsteady analyses.

Figure 7: Eigenvalue trajectories of the dominant modes S1, S2 (solid) and FM (fluid buffet), when altering structural stiffness. (a) growth rate over reduced velocity and (b) frequency over reduced velocity.

ESR15 has analyzed the stability and transition to turbulence in waviness-induced separated flows, contributing to D3.1. ESR15 investigated the instability characteristics and laminar-turbulent transition of a series of laminar separation bubbles (LSBs) formed due to surface waviness, in the absence of external disturbances or forcing. A scaling law based on the geometrical parameters of the waviness and the flow Reynolds number was developed, which enables the prediction of flow separation on the wall leeward side. The analysis of three-dimensional instabilities in two-dimensional baseflows reveals a relationship between the number of changes in the curvature sign of the recirculating streamlines and the number of unstable centrifugal modes that co-exist for the same flow