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We identify a benchmark problem simple enough that it can be solved both by an eddy-viscosity model and by direct numerical simulation: this is the linear response of a turbulent flow’s mean-velocity profile to an external volume force. An example of such a force was found in a study of the perturbation induced by bottom topography by Luchini & Charru (J. Fluid Mech., vol. 656, 2010, pp. 337–341). On the other hand, a modification of the method by Quadrio & Luchini (Proceedings of the IX European Turbulence Conference, Southampton, UK, 2002, pp. 715–718) and Luchini et al. (Phys. Fluids, vol. 18, 2006, 121702) to compute the linear impulse response of a wall-bounded turbulent flow allows the response to a volume force to be computed directly. The comparison exhibits significant differences and suggests that there might be fundamental obstacles to designing an eddy-viscosity model that provides the correct result.