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Polymers injected in a turbulent boundary layer are known to reduce the friction drag by as much as 70%. The complex interaction between polymer chains and turbulent structure has been investigated in the past decade using Direct Numerical Simulations (DNS) and a rheological description based on the Finite Extensible Non-linear Elastic (FENE-Peterlin) constitutive model. Polymers are stretched by the mean shear in the vicinity of the wall and tend to return to their equilibrium just above the buffer layer. Within this process. turbulent kinetic energy is converted (and stored) in elastic energy at the wall, normal fluctuations are strongly reduced and an overall thickening of the viscous region is observed. As the polymers release this excess energy, they tend to reinvigorate the turbulent fluctuations within coherent streamwise structures. Turbulence is not suppressed in polymer drag reduced flows although the usual cycle of energy transfer from the mean flow to the turbulent fluctuations and the anisotropy of the Reynolds stresses are strongly altered. DNS is invaluable in providing insights on the phenomenological process of polymer/turbulence interaction but its application is limited to extremely low Reynolds numbers and simple configurations. To investigate the feasibility of using polymer injection to reduce the friction drag on ships and submarines, an engineering physics-based model has been developed. The starting point is an accurate Reynolds-Averaged Navier-Stokes eddy viscosity closure for Newtonian fluid, the v2-f model [Durbin 1995]. This approach is an extension of the conventional k-e model and accounts for the anisotropy and damping of the wall normal fluctuations in the near-wall region. The addition of polymers in the fluid is modeled mathematically by a set of non-Newtonian stresses added to the Navier-Stokes equations. These are described using the FENEP model: a tensor transport equation (6 equations) is considered to describe the elongation and orientation of the polymer chains. The addition of polymer stresses introduces unclosed terms in the equations for the turbulent scalars; the turbulent kinetic energy equations, for example, is derived directly from the Navier Stokes equations and includes, in non-Newtonian fluids, an extra contribution that describes the energy transfer between polymers and turbulence. The proposed model consists of two parts, the model for the polymer stresses and the closure for the turbulent scalar transport. The polymer stresses are defined in terms of the conformation tensor equations and, in a Reynolds average sense, contain unclosed terms that correlate the turbulent fluctuations to the non-Newtonian stresses. In order to simplify the model, only the polymer elongation is considered, in the assumption that the chains are aligned with the mean strain direction. This allows to consider only one transport equation for the trace of the conformation tensor. The non-Newtnonian stresses are reconstructed using a Boussinesq like assumption, e.g. a linear relationship between mean strain and the polymer stresses. The proportionality coefficient (that plays a role similar to the eddy viscosity for Reynolds stresses) has two contribution, one purely laminar which characterize the response of the polymer to the mean velocity gradient, and the second that accounts for the non-Newtonian stresses supported by the turbulent fluctuations. DNS data suggest that this last contribution is proportional to the turbulent kinetic energy. The second part of the model, involves the closure of the turbulence equations. It has been recognized that the energy redistribution process close to walls is altered by the presence of polymers. Here, a new equilibrium between turbulence production, dissipation and visco-elastic stretching is invoked and this leads to a modification of the pressure-strain term. This formulation naturally introduces enhanced viscous damping at wall and consistently models the elastic energy storage process (in the equation for the polymer elongation) and the release (in the equation for the turbulent kinetic energy). The complete model includes 4 equations for the turbulent stresses (as in the Newtonian case) and 1 equation for the polymer stresses. Initial tests and comparisons provide encouraging results: homogeneous channel flow simulations are compared to DNS data showing good agreement of the velocity profile both for low and high levels of drag reduction.

Polymers are typically injected in turbulent boundary layer through slots or cavities. Experimental observations have shown that they tend to remain very close to the walls, thus producing long-lasting drag reduction even for small injected quantities. Inhomogeneous polymer solutions are challenging to model from the rheological point of view; concentration dependent properties are introduced. In addition, a transport equation for the polymer concentration has to be included in the system. In this case the concentration equation is not “passive” as it directly affects the polymer stresses. DNS data in this case, clearly demonstrate that the turbulent diffusion flux is negligible in the wall-normal direction well above the viscous sublayer. This explains the observed localization of the polymers close to the walls. Conventional models for the turbulent flux are based on a gradient diffusion hypothesis, where the flux is directly proportional to the mean concentration gradient. In this situation, this model leads to considerably high diffusion and cannot be used. A new closure has been developed based on DNS analysis; in this case the flux is directly proportional to the gradient of the polymer stresses. The complete model for inhomogeneous flows has been applied to the flow over a flat plate with injection through a slot.

Calculations have been performed for a range of flow conditions and polymer injection parameters. The friction coefficient and velocity profiles have been compared to experimental data collected at the US Navy Large Cavitation Tunnel. The agreement is satisfactory over a wide range of cases.

In summary, the formulation of the model and the complete set of equations are reported together with the DNS data used for the basic underlying assumptions. In addition, the details of the rheological model used are reported. Calculations carried out for the turbulent flow over a flat plate are reported for three free stream velocities, two injection concentrations and two injection rates. The data are compared to experimental data showing an overall positive agreement. A discussion of the future extensions of the present model is also reported.

Bibliography

Durbin, P., 1995. “Separated Flow Calculations using he k-e-v2 model”, AIAA J., 33, 659, 664. Dubief, Y., Iaccarino, G., Lele, S., 2004. “A Turbulence Model for Polymer Flows”, CTR Annual Research Briefs, 2004.

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