Quantification of model error, i.e. uncertainty associated with modeling assumptions, remains one of the most challenging aspects of uncertainty quantification and predictive simulation. We develop a novel strategy for model error quantification by directly embedding a discrepancy representation into the parameters of the model-to-be-calibrated. The embedded structure is particularly advantageous for predictive science and engineering applications: it enables physically meaningful predictions of quantities of interest (QoIs) by automatically inheriting physical laws and constraints imposed from the model, and provides an intuitive platform for extrapolating the model error for predicting QoIs outside those used for calibration. We characterize and propagate the model error and parameter uncertainties under a Bayesian framework. The calibration problem is solved using approximate likelihood constructions, adaptive Markov chain Monte Carlo, and polynomial chaos expansions. The overall method is demonstrated on large eddy simulation (LES) computations of turbulent flow in a Scramjet engine.