Flow separation is encountered in many engineering devices, e.g., turbines, diffusers, wings and aftbodies of aircrafts. The physical mechanisms of separated turbulent boundary layers over curved walls are not yet well understood. The main objectives of the present study are to: (i) develop an efficient numerical methodology to perform direct numerical simulations (DNS) of spatially-developing turbulent boundary layers (SDTBLs) over curved walls, and (ii) enhance our knowledge on the dynamics of turbulence in SDTBLs separating over curved walls. To achieve these objectives, we have developed a new pressure-correction method, called FastRK3, for simulating incompressible flows over curved walls. FastRK3 solves the incompressible Navier-Stokes (NS) equations written in orthogonal curvilinear coordinates. The orthogonal formulation of the NS equations substantially reduces the computational cost of the flow solver and the numerical stencils of its second-order finite difference discretization mirror that of the Cartesian formulation. This property allows us to develop an FFT-based Poisson solver for pressure, called FastPoc, for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric shapes). Our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver, and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. FastRK3 is an explicit, three-stage, third-order Runge-Kutta based projection-method which requires solving the Poisson equation for pressure only once per time step. We show theoretically and numerically that (i) FastRK3 has the same temporal order of accuracy for pressure and velocity as the standard RK3 method for both free-shear and wall-bounded flows when the RK3 coefficients and the pressure extrapolation scheme satisfy specific conditions herein theoretically derived, (ii) FastRK3 is third-order accurate in time for velocity and second-order accurate in time for pressure for free-shear flows, and (iii) FastRK3 is second-order accurate in time for velocity and pressure for `stiff' wall-bounded flows. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 simulates flows over curved walls with second-order accuracy in both space and time. Using FastRK3, we perform DNS of a SDTBL separating over a curved wall. We validate FastRK3 by comparing our numerical results with published experiments. For the first time, we derive the budget equations of the turbulence kinetic energy and of the Reynolds stresses in orthogonal coordinates, and report the results from our DNS. We study the dynamics of turbulence of the separated flow over the curved wall by analyzing these budget equations. Our analysis shows that, in the separated region over the curved ramp, the TKE production occurs through the production of (u2) as well as (v2) in contrast to a ZPG SDTBL where the TKE production is mostly through the production of (u2). In the curved ramp region, the viscous diffusion and dissipation of (v2) and (uv) are not zero at the wall, unlike that for both a ZPG SDTBL over a flat-plate as well as a pressure-gradient induced turbulent flow separation over a flat plate. And, the curved ramp region of the flow is characterized by enhanced transport of the Reynolds stresses compared to those of the upstream ZPG SDTBL due to the mixing layer created in the flow by the flow separation. Finally, our results have shown, for the first time, that the Reynolds stress profiles and budgets in the orthogonal curvilinear coordinates are very similar to those in the APG region of the 'pressure-gradient induced flow separation' in a flat-plate turbulent boundary layer. Such a comparison is only possible because (i) we employ a structured orthogonal grid over the curved ramp in our simulations, and (ii) FastRK3 solves the governing equations written in orthogonal curvilinear coordinates.