A new simulation approach for high Reynolds number turbulent flows is developed, combining concepts of monotonicity in nonlinear conservation laws with concepts of large-eddy simulation. The spectral vanishing viscosity (SVV), first introduced by E. Tadmor [SIAM J. Numer. Anal. 26, 30 (1989)], is incorporated into the Navier-Stokes equations for controlling high-wavenumber oscillations. Unlike hyperviscosity kernels, the SVV approach involves a second-order operator which can be readily implemented in standard finite element codes. In the work presented here, discretization is performed using hierarchical spectral/hp methods accommodating effectively an ab initio intrinsic scale separation. The key result is that monotonicity is enforced via SVV leading to stable discretizations without sacrificing the formal accuracy, i.e., exponential convergence, in the proposed discretization. Several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy-viscosity spectral LES of turbulent channel flow. In its current implementation the SVV approach for controlling the small scales is decoupled from the large scales, but a procedure is proposed that will provide coupling similar to the classical LES formulation.