Simulations of combustion in high-speed and supersonic flows need to account for autoignition phenomena, compressibility, and the effects of intense turbulence. In the present work, the evolution-variable manifold framework of Cymbalist and Dimotakis ("On Autoignition-Dominated Supersonic Combustion," AIAA Paper 2015-2315, June 2015) is implemented in a computational fluid dynamics method, and Reynolds-averaged Navier-Stokes and wall-modeled large-eddy simulations are performed for a hydrogen-air combustion test case. As implemented here, the evolution-variable manifold approach solves a scalar conservation equation for a reaction-evolution variable that represents both the induction and subsequent oxidation phases of combustion. The detailed thermochemical state of the reacting fluid is tabulated as a low-dimensional manifold as a function of density, energy, mixture fraction, and the evolution variable. A numerical flux function consistent with local thermodynamic processes is developed, and the approach for coupling the computational fluid dynamics to the evolution-variable manifold table is discussed. Wall-modeled large-eddy simulations incorporating the evolution-variable manifold framework are found to be in good agreement with full chemical kinetics model simulations and the jet in supersonic crossflow hydrogen-air experiments of Gamba and Mungal ("Ignition, Flame Structure and Near-Wall Burning in Transverse Hydrogen Jets in Supersonic Crossflow," Journal of Fluid Mechanics, Vol. 780, Oct. 2015, pp. 226-273). In particular, the evolutionvariable manifold approach captures both thin reaction fronts and distributed reaction-zone combustion that dominate high-speed turbulent combustion flows.Nomenclature a = speed of sound, m∕s C; X ; Z = progress variable, nonfuel mass fraction, and fuel mass fraction c v ; c v;s = mixture and s-species specific heats, J∕kg ⋅ K D = diffusion coefficient, m 2 ∕s E = total energy per unit volume, J∕m 3 e; e s = mixture and s-species specific energy, J∕kg F = convective flux vector h; h 0 = enthalpy and total enthalpy, J∕kg i = grid index J = jet momentum ratio j h = diffusive enthalpy flux k = kinetic energy, J∕kg L; R = values obtained from left and right data M s = s-species molar mass, kg∕kmol N s = number of species in detailed kinetics model n; n x ; n y ; n z = element face unit normal vector and components p = pressure, Pa q j = heat flux vector= vectors of conserved and primitive variables u; u; v; w = velocity vector and components, m∕s u 0 = face-normal velocity component, m∕s x; x j = position vector and its components, m Y s = s-species mass fraction α = dissipative flux factor δ R = reaction-zone thickness, m ε = dissipation rate ζ = evolution-variable source term, 1∕s η k = Kolmogorov length scale, m Λ; λ = diagonal matrix of eigenvalues and eigenvalue ν t ; ν = turbulence field variable and kinematic viscosity ρ; ρ s = density and s-species density, kg∕m 3 σ ij = viscous and Reynolds stress τ = evolution variable ϕ = stoichiometric fuel-air ratio χ = subgrid-s...